Computing discrete equivariant harmonic maps
CComputing discrete equivariant harmonic maps
Jonah Gaster ∗ , Brice Loustau † , and Léonard Monsaingeon ‡ Abstract
We present effective methods to compute equivariant harmonic maps from the universalcover of a surface into a nonpositively curved space. By discretizing the theory appropriately,we show that the energy functional is strongly convex and derive convergence of the discreteheat flow to the energy minimizer, with explicit convergence rate. We also examine centerof mass methods, after showing a generalized mean value property for harmonic maps.We feature a concrete illustration of these methods with
Harmony , a computer softwarethat we developed in C ++ , whose main functionality is to numerically compute and displayequivariant harmonic maps. Key words and phrases:
Harmonic maps · Heat flow · Convexity · Gradient descent · Centers of mass · Discrete geometry · Riemannian optimization · Mathematical software
Primary: 58E20; Secondary: 53C43 · ∗ McGill University, Department of Mathematics and Statistics. Montreal, QC H3A 0B9, Canada.E-mail: [email protected] † Rutgers University - Newark, Department of Mathematics. Newark, NJ 07105 USA; and TU Darmstadt,Department of Mathematics. 64289 Darmstadt, Germany. E-mail: [email protected] ‡ IECL Université de Lorraine, Site de Nancy. F-54506 Vandœuvre-lès-Nancy Cedex, France; and GFM Univer-sidade de Lisboa. 1749-016 Lisboa, Portugal. E-mail: [email protected] a r X i v : . [ m a t h . G T ] J a n ontents Introduction 3Acknowledgments 51 Harmonic maps 6 ( M , N ) and more general spaces . . . . . . . . . . . . . . 71.3 The heat flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Equivariant harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Harmonic maps from surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 H . . . . . . . . . . . . . . . . . . . . 233.5 More general target spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Harmony Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References 51 ntroduction
The theory of harmonic maps has its roots in the foundations of Riemannian geometry and theessential work of Euler, Gauss, Lagrange, and Jacobi. It includes the study of real-valued harmonicfunctions, geodesics, minimal surfaces, and holomorphic maps between Kähler manifolds.A harmonic map f : M → N is a critical point of the energy functional E ( f ) = ∫ M (cid:107) d f (cid:107) d v M . The theory of harmonic maps was brought into a modern context for Riemannian manifoldswith the seminal work of Eells-Sampson [ES64] (also Hartman [Har67] and Al’ber [Al’68]). Eells-Sampson studied the heat flow associated to the energy, i.e. the nonlinear parabolic PDEdd t f t = τ ( f t ) , where τ ( f ) is the tension field of f (see § 1). The tension field can be described as minus thegradient of the energy functional on the infinite-dimensional Riemannian space C ∞ ( M , N ) , so thatthe heat flow is just the gradient flow for the energy. When N is compact and nonpositively curved,the heat flow is shown to converge to an energy-minimizing map as t → ∞ .The theory has since been developed and generalized to various settings where the domain or thetarget are not smooth manifolds [GS92, KS93, Che95, Jos97, EF01, Mes02, DM08]. In particular,Korevaar-Schoen developed an extensive Sobolev theory when the domain is Riemannian but thetarget is a nonpositively curved metric space [KS93, KS97]. Jost generalized further to a domain thatis merely a measure space [Jos84, Jos94, Jos95, Jos96]. Both took a similar approach, constructingthe energy functional E as a limit of approximate energy functionals.These tools have become powerful and widely used, with celebrated rigidity results [Siu80,GS92, Wan00, DMV11, IN05] and dramatic implications for the study of deformation spaces andTeichmüller theory when the domain M is a surface (see e.g. [DW07]), especially via the nonabelianHodge correspondence [Don87, Hit87, Cor88, Wol89, Lab91, Sim91].This paper and its sequel [GLM19] are concerned with effectiveness of methods for findingharmonic maps. In addition to the mathematical content, we feature Harmony , a computer programthat we developed in C ++ whose main functionality is to numerically compute equivariant harmonicmaps. Our project was motivated by the question: is it possible to study the nonabelian Hodgecorrespondence experimentally? Though the heat flow is constructive to some extent, it does notprovide qualitative information about convergence in general. We show that one can design entirelyeffective methods to compute harmonic maps by discretizing appropriately.Some of the existing literature treats similar questions, though seldom in the Riemanniansetting. Notably, Bartels applies finite element methods on submanifolds of R n to nonlinear PDEssuch as the Euler-Lagrange equations for minimizing the energy [Bar05, Bar10, Bar15]. There isalso an extensive literature on discrete energy functionals of the form we consider in this paper[CdV91, PP93, Leb96, KS01, JT07, Wan00, IN05, EF01, Fug08, HS15].However, several features set our work apart. Firstly, we study sequences of arbitrarily finediscretizations as opposed to one fixed approximation of the manifold, particularly in the sequelpaper [GLM19]. Moreover, we follow the maxim of Bobenko-Suris [BS08, p. xiv]: Discretize the whole theory, not just the equations .Our discrete structures record two independent systems of weights on a given triangulation of thedomain manifold M , on the set of vertices and on the set of edges respectively. The vertex weights are3 discrete record of the volume form, while the edge weights generalize the well-known “cotangentweights” popularized by Pinkall-Polthier in the Euclidean setting [PP93]. In the 2-dimensionalcase, the edge weights can be thought of as a discrete record of the conformal structure. Thisdiscretization endows the space of discrete maps with a finite-dimensional Riemannian structure,approximating the L metric on C ∞ ( M , N ) . We obtain the right setting for a study of the convexityof the discrete energy, and for the definitions of the discrete energy density, discrete tension field,and discrete heat flow. Among the practical benefits, we find that the discrete energy satisfiesstronger convexity properties than those known to hold in the smooth setting.The theory of harmonic maps is also intimately related to centers of mass. In the Euclideansetting, harmonic functions satisfy the well-known mean value property. While the latter no longerholds stricto sensu in the more general Riemannian setting, we prove a generalization in terms ofcenters of mass of harmonic maps on very small balls. More generally, averaging a function onsmall balls offers a viable method in order to decrease its energy, a viewpoint well adapted to Jost’stheory of generalized harmonic maps [Jos94, Jos95, Jos96, Jos97]. As an alternative to the discreteheat flow, we pursue a discretization of the theory of Jost by analyzing discrete center of massmethods.For both heat flow and center of mass methods, the present paper focuses on a fixed discretizationof the domain, while the sequel paper [GLM19] analyzes convergence of the discrete theory backto the smooth one as we take finer and finer discretizations approximating a smooth domain. Whilethis paper focuses on two dimensional hyperbolic manifolds, and chooses the equivariant setting, thesecond paper embraces the general Riemannian setting for the most part, and drops the equivariantformulation.Now let us describe more precisely some of the main theorems of the paper. After discussingharmonic maps in § 1 and developing a discretized theory in § 2, we study the convexity of theenergy functionals in § 3. We show: Theorem (Theorem 3.20) . Let S be a discretized surface of negative Euler characteristic and let N be a compact surface of nonzero Euler characteristic with nonpositive sectional curvature. Thediscrete energy functional is strongly convex in any homotopy class of nonzero degree. We stress that while the convexity of the energy comes for free, the content of the theorem aboveis a positive modulus of convexity, i.e. a uniform lower bound for the Hessian. We actually showa more general version of this theorem involving equivariant maps and the notion of biweightedtriangulated graph which we introduce in § 2. See Theorem 3.20 for the precise statement.When the target N is specialized to a hyperbolic surface, we find explicit bounds for the Hessianof the energy functional (see Theorem 3.18). We achieve this through detailed calculations in thehyperbolic plane, which we then generalize to negatively curved target surfaces using CAT ( k ) -typecomparisons. Roughly speaking, the key idea is that if the energy of some function has a very smallsecond variation, then one can construct an almost parallel vector field on the image of the surface;however this is not possible for topological reasons. § 3 is concerned with the significant work ofmaking this argument precise and quantitative.In § 4 we study gradient descent methods in Riemannian manifolds and specialize to theconvergence of the discrete heat flow. Fully leveraging strong convexity of the energy, we show: Theorem (Theorem 4.5) . Let S be a discretized surface of negative Euler characteristic and let N be a compact surface of nonzero Euler characteristic with nonpositive curvature. There exists aunique discrete harmonic map f ∗ : S → N in any homotopy class of nonzero degree. Moreover,for any initial value f : S → N and for any sufficiently small t > , the discrete heat flow withfixed stepsize t converges to f ∗ with exponential convergence rate. f on balls of small radius r >
0, producing a new map B r f : M → N . Repeatingthis process potentially produces energy-minimizing sequences for an approximate version of theenergy E r , a phenomenon that has been explored by Jost [Jos94]. The central theorem we prove in§ 5.2 is that in the Riemannian setting, this iterative process is almost the same as a fixed stepsizetime-discretization of the heat flow. See Theorem 5.8 for a precise statement.We prove in particular the following generalized mean value property for harmonic maps: Theorem (Theorem 5.9) . Let f : M → N be a smooth map between Riemannian manifolds. Then f is harmonic if and only if for all x ∈ M , we have as r → : d ( f ( x ) , B r f ( x )) = O ( r ) . Under suitable conditions, we show that the center of mass method converges to a minimizerof the approximate energy E r (see Theorem 5.17), recovering a theorem of Jost [Jos94, §3]. Jost’sresult is more general, but our conclusion is slightly stronger.In the space-discretized setting, the approximate energy coincides with the discrete energy,making the discrete center of mass method an appropriate alternative to the discrete heat flow. Theorem (Theorem 5.21) . Let S be a discretized surfaceand N be a compact manifold of negativesectional curvature. In any π -injective homotopy class of maps S → N , the center of mass methodfrom any initial map converges to the unique discrete harmonic map. As a concrete demonstration of the effectiveness of our algorithms, we present in § 6 ourown computer implementation of a harmonic map solver:
Harmony is a freely available computersoftware with a graphical user interface written in C ++ code, using the Qt framework. Thisprogram takes as input the Fenchel-Nielsen coordinates for a pair of Fuchsian representations ρ L , ρ R : π S → Isom ( H ) and computes and visualizes the unique equivariant harmonic map. Harmony ’s main user interface is illustrated in Figure 6.In future development of
Harmony , we plan to compute and visualize harmonic maps for moregeneral target representations that are not necessarily discrete, and for more general target spaces,such as H and other nonpositively curved symmetric spaces.We have implemented both the discrete heat flow method, with fixed and optimal stepsizesseparately, and the cosh-center of mass method, a clever variant of the center of mass methodsuggested to us by Nicolas Tholozan that is better suited for computations in hyperbolic space(discussed in § 5.4). In practice, the cosh-center of mass method is the most effective, both innumber of iterations and execution time (see § 4.3). Acknowledgments
The authors wish to thank Tarik Aougab, Benjamin Beeker, Alexander Bobenko, David Dumas, JohnLoftin, Jean-Marc Schlenker, Nicolas Tholozan, and Richard Wentworth for valuable conversationsand correspondences pertaining to this work. We especially thank David Dumas for his extensiveadvice and support with the mathematical content and the development of
Harmony , and NicolasTholozan for sharing key ideas that contributed to Theorem 3.20 and Theorem 5.9.The first two authors gratefully acknowledge research support from the NSF Grant DMS1107367
RNMS: GEometric structures And Representation varieties (the
GEAR Network ). The third authorwas partially supported by the Portuguese Science Foundation FCT trough grant PTDC/MAT-STA/0975/2014
From Stochastic Geometric Mechanics to Mass Transportation Problems .5 Harmonic maps
Let ( M , g ) and ( N , h ) be two smooth Riemannian manifolds. Assuming M is compact, the energy of a smooth map f : M → N is: E ( f ) = ∫ M (cid:107) d f (cid:107) d v g (1)where v g is the volume density of the metric g . Note that d f is a smooth section of the bundleT ∗ M ⊗ f ∗ T N over M , which admits a natural metric induced by g and h , giving sense to (cid:107) d f (cid:107) .This is the so-called Hilbert-Schmidt norm of d f , which is also described as (cid:107) d f (cid:107) = tr g ( f ∗ h ) . Definition 1.1.
A map f : M → N is harmonic if it is a critical point of the energy functional (1).This means concretely that: dd t | t = E ( f t ) = ( f t ) : (− δ, δ ) × M → N of f = f . Note that one should work withcompactly supported deformations when M is not compact, as the energy could be infinite.A more tangible characterization of harmonicity is given by the Euler-Lagrange equation for E ,which takes the form τ ( f ) = τ ( f ) is the tension field of f : this is an immediate consequenceof the first variational formula below (Proposition 1.3). First we define the tension field. Note thatthe bundle T ∗ M ⊗ f ∗ T N admits a natural connection ∇ induced by the Levi-Civita connections of g and h . Hence one can take the covariant derivative ∇( d f ) ∈ Γ ( T ∗ M ⊗ T ∗ M ⊗ f ∗ T N ) (we use thenotation Γ for the space of smooth sections), also denoted ∇ f . It is easily shown to be symmetricin the first two factors. Definition 1.2.
The vector-valued Hessian of f is ∇ f (cid:66) ∇( d f ) ∈ Γ ( T ∗ M ⊗ T ∗ M ⊗ f ∗ T N ) . The contraction (trace) of ∇ f on its first two indices using the metric g is the tension field of f : τ ( f ) (cid:66) tr g (∇ f ) ∈ Γ ( f ∗ T N ) . Note that the vector-valued Hessian generalizes both the usual Hessian (when N = R ) and the(vector-valued) second fundamental form (when f is an isometric immersion). Accordingly, thetension field generalizes both the Laplace-Beltrami operator and the (vector-valued) mean curvature. Proposition 1.3 (First variational formula for the energy) . Let f : ( M , g ) → ( N , h ) be a smooth mapand let ( f t ) be a smooth deformation of f . Denote by V ∈ Γ ( f ∗ T N ) the associated infinitesimaldeformation defined as V x = dd t | t = f t ( x ) . Then dd t | t = E ( f t ) = − ∫ M (cid:104) τ ( f ) x , V x (cid:105) d v g ( x ) (2) where h = (cid:104)· , ·(cid:105) is the Riemannian metric in N . inner product of two infinitesimal deformations V , W ∈ Γ ( f ∗ T N ) (also called vector fields along f ): (cid:104) V , W (cid:105) = ∫ M (cid:104) V x , W x (cid:105) d v g ( x ) . (3)There is in fact a natural smooth structure on C ∞ ( M , N ) , making it an infinite-dimensional manifold,which identifies the tangent space at f asT f C ∞ ( M , N ) = Γ ( f ∗ T N ) , (4)we refer to [KM97, Chapter IX] for details. With respect to this smooth structure, (3) defines aRiemannian metric on C ∞ ( M , N ) , and (2) can simply be put:grad E ( f ) = − τ ( f ) . (5)Next we compute the second variation of the energy (like the first variation, this is already in[ES64]): Proposition 1.4 (Second variational formula for the energy) . Let ( f st ) : (− δ, δ ) × M → N be asmooth deformation of f = f . Denote V = ∂ f ∂ s | s = and W = ∂ f ∂ t | t = . Then ∂ E ( f st ) ∂ s ∂ t | s = t = = ∫ M (cid:18) (cid:104)∇ V , ∇ W (cid:105) − tr g (cid:10) R N ( d f , V ) W , d f (cid:11) + (cid:28) ∇ ∂∂ t ∂ f ∂ s , τ ( f ) (cid:29)(cid:19) d v g where R N is the Riemann curvature tensor on N . When ( f st ) is a geodesic variation, i.e. f st ( x ) = exp f ( x ) ( sV x + tW x ) , the third term in the integralvanishes. This yields the formula for the Hessian of the energy functional:Hess ( E ) | f ( V , W ) = ∫ M (cid:16) (cid:104)∇ V , ∇ W (cid:105) − tr g (cid:10) R N ( V , d f ) d f , W (cid:11)(cid:17) d v g . (6)When M is closed, this can also be written Hess ( E ) | f ( V , W ) = (cid:104) J ( V ) , W (cid:105) using the L Riemannianmetric (3), where J ( V ) = − tr g (∇ V + R N ( V , d f ) d f ) is the Jacobi operator . L ( M , N ) and more general spaces The energy functional can be extended to maps that are merely in L ( M , N ) . First let us definethis function space. Assume M is compact. The L -distance between two measurable maps f , f : M → N is d ( f , f ) = (cid:18)∫ M d ( f ( x ) , f ( x )) d v g ( x ) (cid:19) . (7)If f , f are both smooth, this is the distance induced by the L Riemannian metric (3), providedthere exists a geodesic between f and f . A measurable map f : M → N is declared in L ( M , N ) when it is within finite distance of a constant map. For r >
0, one can then define an approximate r -energy of f ∈ L ( M , N ) : E r ( f ) = ∫ M ∫ M η r ( x , y ) d ( f ( x ) , f ( y )) d v g ( y ) d v g ( x ) (8) For us the curvature tensor is R ( X , Y ) Z = ∇ X , Y Z − ∇ Y , X Z . Some authors’ convention differs in sign, e.g. [GHL04]. η r ( x , y ) is a kernel that may be chosen η r ( x , y ) = r ( x , y ) r V m ( r ) , where V m ( r ) is the volume of a ballof radius r in a Euclidean space of dimension m = dim M and r ( x , y ) is the characteristic functionof {( x , y ) ∈ M : d ( x , y ) < r } in M × M (see [Jos97, §4.1] for a discussion of the choice of kernel).One can show that the functional E r is continuous on L ( M , N ) . Moreover, the limit: E ( f ) (cid:66) lim r → E r ( f ) (9)exists in [ , ∞] for every f ∈ L ( M , N ) . The resulting energy functional E is lower semi-continuouson L ( M , N ) and coincides with (1) on C ∞ ( M , N ) . A measurable map f : M → N is declared inthe Sobolev space H ( M , N ) if it is in L ( M , N ) and has finite energy. The spaces L ( M , N ) andH ( M , N ) are similarly defined by restricting to compact sets. One can then define a (weakly)harmonic map as a critical point of the energy functional in H ( M , N ) . Any continuous weaklyharmonic map is smooth [Jos17, Theorem 9.4.1] (the continuity assumption can be dropped when M and N are compact and N has nonpositive curvature: [Jos17, Corollary 9.6.1]).In addition to opening the way for tools from functional analysis, this approach can be generalizedto much more general spaces than Riemannian manifolds. Indeed, assume M = ( M , µ ) is a measurespace and N = ( N , d ) is a metric space. The space L ( M , N ) may be defined as before, and givena choice of kernel η r for r >
0, one can define energy functionals E r : L ( M , N ) → R using (8).For a suitable choice of η r and of a sequence r n →
0, the energy functional is E = lim n → + ∞ E r n .More precisely, one must ensure that E is the Γ -limit of the functionals E r n . We refer to [Jos97,Chap. 4] for details and [DM93] for the theory of Γ -convergence. Γ -convergence is adequate herebecause it ensures that minimizers of E r converge to minimizers of E . This point of view on thetheory of harmonic maps was developed by Jost [Jos94, Jos95, Jos96, Jos97]. A similar approachwas developed by Korevaar-Schoen [KS93, KS97]. Going back to the smooth setting, assume that M is compact and N is complete and has nonpositivecurvature. The formula for the Hessian of the energy (6) shows that it is nonnegative, in otherwords E is a convex function on C ∞ ( M , N ) with respect to the L Riemannian metric. Thismakes it reasonable to expect existence and in certain cases uniqueness of harmonic maps, whichare necessarily energy-minimizing in this setting (we discuss this further in § 3.2). A naturalapproach to minimize a convex function is the gradient flow, called heat flow in this setting: given f ∈ C ∞ ( M , N ) , consider the initial value problem dd t f t = − grad E ( f t ) , that is in light of (5):dd t f t = τ ( f t ) . This flow exists for all t (cid:62)
0. Moreover, if the range of f t remains in some fixed compact subset of N ,then f t converges to a harmonic map as t → ∞ , uniformly and in L ( M , N ) (in fact, in C ∞ ( M , N ) ).Otherwise, there exists no harmonic map homotopic to f . In particular, when N is compact, any f ∈ C ∞ ( M , N ) is homotopic to a smooth energy-minimizing harmonic map. Moreover, such aharmonic map is unique, unless it is constant or maps into a totally geodesic flat submanifold of N in which case non-uniqueness is realized by translating f in the flat. These foundational results aredue to Eells-Sampson [ES64] and Hartman [Har67].8 .4 Equivariant harmonic maps Instead of working with maps between compact manifolds, it can be useful to study their equivariantlifts to the universal covers. Indeed, up to being careful with basepoints, any continuous map f : M → N lifts to a unique ρ -equivariant map ˜ f : ˜ M → ˜ N , where ρ : π M → π N is the grouphomomorphism induced by f . Note that ρ only depends on the homotopy class of f , and if N isaspherical (i.e. ˜ N is contractible), then conversely any ρ -equivariant continuous map M → N is thelift of some continuous map M → N homotopic to f .This approach enables the following generalization: let X and Y be two Riemannian manifolds,denote Isom ( X ) and Isom ( Y ) their groups of isometries. Let Γ be a discrete group. Given grouphomomorphisms ρ L : Γ → Isom ( X ) and ρ R : Γ → Isom ( Y ) , a map f : X → Y is called ( ρ L , ρ R ) -equivariant if: f ◦ ρ L ( γ ) = ρ R ( γ ) ◦ f for all γ ∈ Γ . Note that the quotients X / ρ L ( Γ ) and Y / ρ R ( Γ ) can be pathological, but the space ofequivariant maps X → Y remains ripe for study.The heat flow approach of Eells-Sampson to show existence of harmonic maps between compactRiemannian manifolds when the target is nonpositively curved has been successfully adapted tothe equivariant setting by various authors. The adequate condition for guaranteeing existence ofequivariant harmonic maps is the reductivity of the target representation. More precisely: Theorem 1.5 ([Lab91]) . Let M and N be Riemannian manifolds, assume N is Hadamard. Denoteby ρ L : π M → Isom ( ˜ M ) the action by deck transformations and let ρ R : π M → Isom ( N ) be anygroup homomorphism. If ρ R is reductive, then there exists a ( ρ L , ρ R ) -equivariant harmonic map ˜ M → N . The converse also holds provided N is without flat half-strips. Less general versions of this theorem had previously been established by Donaldson [Don87](for N = H ) and Corlette [Cor88] (for N a Riemannian symmetric space of noncompact type). Thenotion of being reductive for a group homomorphism ρ : π M → G can be described algebraicallywhen N = G / K is a Riemannian symmetric space of noncompact type . Labourie [Lab91]generalized it to Hadamard manifolds. When N has negative curvature, ρ is reductive if and onlyif it fixes no point on the Gromov boundary ∂ ∞ N or it preserves a geodesic in N .Wang [Wan00] and Izeki-Nayatani [IN05] generalized Theorem 1.5 to Hadamard metric spacesusing Jost’s extended notion of reductivity [Jos97, Def. 4.2.1]. Less general or different versionswere previously established by [GS92], [Jos97, Thm 4.2.1], [KS97]. When M = S is a surface, i.e. dim M =
2, it is easy to check that the energy density element e ( f ) d v g (cid:66) (cid:107) d f (cid:107) d v g is invariant under conformal changes of the metric g . Thus the energyfunctional only depends on the conformal class of g , as does the harmonicity of a map S → N . Aconformal structure on an oriented surface is equivalent to a complex structure (this follows froma result going back to Gauss [Gau25] on the existence of conformal coordinates). Hence one maytalk about the energy and harmonicity of maps X → ( N , h ) where X is a Riemann surface. Notehowever that the L metric (3) does change under conformal changes of g , therefore the tensionfield τ ( f ) does too, as does the modulus of strong convexity of the energy (see § 3.1). When G is an algebraic group, a subgroup H ⊆ G is completely reducible if, for every parabolic subgroup P ⊆ G containing H , there is a Levi subgroup of P containing H . Equivalently, the identity component of the algebraic closureof H is a reductive subgroup (with trivial unipotent radical). A G -valued group homomorphism ρ is called reductive (orcompletely reducible) when its image is a completely reducible subgroup. Refer to [Sik12] for details. X on S by writing the pullback metric f ∗ h on X . Splitting it into types, one finds that f ∗ h = ϕ f + g f + ¯ ϕ f where ϕ f = ( f ∗ h ) ( , ) is a complex quadratic differential on X , called the Hopf differential of f , and g f = ( f ∗ h ) ( , ) is e ( f ) (more precisely, g f is the conformal metric with volume density e ( f ) d v g ).The Hopf differential ϕ f plays an important role in Teichmüller theory. First note that f isconformal if and only if ϕ f =
0. A key fact is that if f is harmonic, then ϕ f is a holomorphicquadratic differential on X . Wolf [Wol89] proved that the Teichmüller space of X is diffeomorphicto the vector space of holomorphic quadratic differentials on X by taking Hopf differentials ofharmonic maps X → ( S , h ) , where h is a hyperbolic metric on S (see § 6.3) . We refer to [DW07]for a beautiful review of the connections between harmonic maps and Teichmüller theory.On a closed surface S of negative Euler characteristic, it is convenient to choose the Poincarémetric within a conformal class of metrics: it is the unique metric of constant curvature − uniformization theorem ). This provides an identification of ˜ S with the hyperbolic plane H and an action of π S on H by isometries. Turning this identificationaround, whenever a Fuchsian (i.e. faithful and discrete) representation ρ L : π S → Isom + ( H ) ischosen, we obtain a hyperbolic surface H / ρ L ( π S ) ≈ S . We fix some notation: for the remainder of the paper, S is a smooth, closed, oriented surface ofnegative Euler characteristic (genus (cid:62) π S the fundamental group of S with respectto some basepoint that can be safely ignored. Remark . While this paper specializes the discretization to 2-dimensional hyperbolic surfaces,most of the definitions can be generalized to higher-dimensional Riemannian manifolds. The sequelpaper [GLM19] treats the general Riemannian setting (while dropping the equivariant formulation,mostly for comfort).In § 2.1 we explain how to approach the energy minimization problem for smooth equivariantmaps H → N in order to allow effective computation by introducing meshes and subdivisions,discrete equivariant maps, and discrete energy. Several of these notions are further discussed in[GLM19]. In the present paper they serve as a preamble to the more formal setting we develop in§ 2.2 and they justify the choices made in the software Harmony . Let us fix a hyperbolic structure on S given by a Fuchsian representation ρ L , i.e. an injective grouphomomorphism π S → Isom + ( H ) with discrete image. This setup can be easily generalized to anyRiemannian metric on S , but the hyperbolic metric is best suited for computations. Meshes and subdivisions
Given a group homomorphism ρ R : π S → Isom ( N ) where N is a Riemannian manifold (or a metricspace), we would like to discretize ( ρ L , ρ R ) -equivariant maps H → N . To this end, we start bydiscretizing the domain hyperbolic surface with the notion of invariant mesh:10 a) An invariant mesh of H (b) Refinement of order 1 Figure 1: An invariant mesh of the Poincaré disk model of H on the left, its refinement of order 1on the right. The brighter central region is a fundamental domain. The blue circle arcs are the axesof the generators of ρ L ( π S ) . Pictures generated by Harmony . Definition 2.2. A ρ L -invariant mesh of H is an embedded graph M in H such that:(i) The vertex set M ( ) ⊂ H (set of mesh points ) is invariant under a cofinite action of ρ L ( π S ) .(ii) Every edge e ∈ M ( ) is an embedded geodesic segment in H .(iii) The complementary components are triangles.For the purpose of approximating smooth maps, we will need to take finer and finer meshes. Thiswill be discussed in detail in [GLM19], but let us describe the strategy that we have implementedin the software Harmony . A natural way to obtain a finer mesh from a given one is via geodesicsubdivision. We indicate below an edge of M with endpoints x , y ∈ M ( ) by e xy , and let m ( x , y ) ∈ H be the midpoint of x and y . It is easy to see that the following is well-defined: Definition 2.3.
The refinement of a ρ L -invariant mesh M is the ρ L -invariant mesh M (cid:48) such that:(i) The vertices of M (cid:48) are the vertices of M plus all midpoints of edges of M .(ii) The edges of M (cid:48) are given by x ∼ m ( x , y ) and y ∼ m ( x , y ) for each edge e xy , and m ( x , y ) ∼ m ( x , z ) for each triple of vertices x , y , z that span a triangle in M .Evidently, this refinement may be iterated. See Figure 1 for an illustration of a ρ L -invariantmesh and its refinement generated by the software Harmony . Discrete equivariant maps
Given a ρ L -invariant mesh M and a group homomorphism ρ R : π S → Isom ( N ) , we call discreteequivariant map H → N along M a ( ρ L , ρ R ) -equivariant map from the vertex set M ( ) to N .We denote Map M ( H , N ) the set of discrete equivariant maps H → N along M . Note thatMap M ( H , N ) ≈ N V where V = M ( ) / ρ L ( π S ) is the set of equivalence classes of meshpoints,which is finite. Therefore: Proposition 2.4. If N is a finite-dimensional smooth manifold, then so is Map M ( H , N ) . C ( H , N ) the space of continuous equivariant maps H → N , the forgetful map C ( H , N ) → Map M ( H , N ) (10)consists in restricting a continuous function to the set of meshpoints M ( ) . Definition 2.5.
We shall call a right inverse of the forgetful map (10) an interpolation scheme .While there is one most natural way to interpolate discrete maps between Euclidean spaces(affine interpolation), there is no preferred way for arbitrary Riemannian manifolds. Even in the casewhere both domain and target manifolds are the hyperbolic plane H , there are several reasonableinterpolations to consider such as the barycentric interpolation and the harmonic interpolation.However, these are not explicit, and with Harmony we work with a neat variant, the cosh-center ofmass interpolation (see § 5.4).
Discretization of energy
For a smooth ( ρ L , ρ R ) -equivariant map f : H → N where N is a Riemannian manifold, one definesthe total energy of f as: E ( f ) = ∫ D (cid:107) d f (cid:107) d v g (11)where D ⊂ H is any fundamental domain for the action of π S . If D is picked so that it coincideswith a union of triangles defined by M , then the energy can be written as a finite sum of energyintegrals over each triangle. When f is discretized along M , only the values of f on the meshpointsare recorded. Thus, a natural discretization of E is obtained if one knows how to define the energyof a map from a triangle whose values are only known at the vertices. Given an interpolationscheme (cf Definition 2.5), one can simply take the energy of the interpolated map.Another approach consists in defining the energy à la Jost / Korevaar-Schoen as in § 1.2. Onecan take the graph defined by M with metric induced from H as a domain metric space, introducea measure that approximates the area density of H , and choose an appropriate kernel η ( x , y ) .A third approach consists in choosing a discrete energy that is a weighted sum of distancessquared as in Definition 2.6 and Definition 2.12. This approach provides a natural extension of theclassical notion of real-valued harmonic functions defined on graphs [BH12, Chu97, GR01], thatis, functions whose value at any vertex is the average of the values on the neighbors.It turns out that all three approaches can be made to coincide (Proposition 2.19), or almostcoincide for fine meshes, for the appropriate choices involved in the different definitions. This isthoroughly discussed in the sequel paper [GLM19]. Definition 2.6.
Let M be a ρ L -invariant mesh in H such that all the complementary triangles areacute. The discrete energy of a discrete equivariant map f ∈ Map M ( H , N ) is defined by E M ( f ) = (cid:213) e = e xy ∈E ω xy d ( f ( x ) , f ( y )) where:• E ⊂ M ( ) is any fundamental domain for the action of π S on the set of edges.• Inside the sum, x and y are the vertices connected by the edge e .• ω xy is the half-sum of the cotangents of two angles: one for each of the two triangles sharingthe edge e = e xy , in which we take the angle of the vertex facing the edge e .12his definition is a generalization of the energy considered by Pinkall-Polthier [PP93], for whomthe domain is a triangulated surface with a piecewise Euclidean metric and N = R n . In their setting,the discrete energy coincides with the energy relative to the linear interpolation. In [GLM19] weshow that the discrete energy E M converges to the smooth energy E under iterated refinement ofthe mesh M in a suitable function space.Of course, we can now define a discrete equivariant harmonic map f ∈ Map M ( H , N ) as acritical point of the discrete energy functional E M . While several authors have shown the existenceand uniqueness of minimizers of the discrete energy in various contexts (e.g. [Wan00, EF01,Mes02]), in this paper we analyze its strong convexity, which makes it better suited for effectiveminimization. Our approach requires a Riemannian metric on Map M ( H , N ) (cf. § 3.1), whichshould approach the L Riemannian metric of C ∞ ( M , N ) (cf. § 1.1). In the next subsection, wedevelop a more general framework where these ideas apply. The definition of the discrete energy functional E M (Definition 2.6) is easily generalized to anysystem of positive weights indexed by the edges of M . On the other hand, the Riemannian structureof Map M ( H , N ) requires a measure on the domain: while in the smooth case one has the volumedensity of the Riemannian metric, in the discrete case it can be recorded by a system of weights onthe vertices. All of this information can be captured using only the graph structure of M .˜ S -triangulated graphs Recall that a triangulation T of a surface is the data of a simplicial complex K and a homeomorphism h from K to the surface. Lifting T to the universal cover, we find a triangulation whose underlyinggraph G (i.e. 1-skeleton) is locally cyclic , meaning that the open neighborhood of any vertex(subgraph induced on the neighbors) is a cycle . This motivates the following definition: Definition 2.7.
Given a topological surface S , an ˜ S -triangulated graph is a locally cyclic graph G with a free, cofinite action of π S by graph automorphisms.˜ S -triangulated graphs are precisely the graphs that arise as 1-skeleta of triangulations. When G is an ˜ S -triangulated graph, we denote the associated group action on the set of vertices by ρ L : π S → Aut (G ( ) ) . Let N be a metric space and ρ R : π S → Isom ( N ) a group homomorphism. Definition 2.8.
Given ρ L and ρ R as above, we call a ( ρ L , ρ R ) -equivariant map G ( ) → N an equivariant map from G to N . The space of such equivariant maps will be denoted Map eq (G , N ) .As in Proposition 2.4 we have: Proposition 2.9. If N is a finite-dimensional smooth manifold, so is Map eq (G , N ) . Note that G has the property that every face (2-simplex) of the triangulation is a triangle (3-vertex completesubgraph) in G ; when the converse is also true one says that T is a Whitney triangulation . In other words, a Whitneytriangulation can be recovered as the flag complex spanned by G . Let us cite [LNLPn02] here: Whitney triangulationsare quite amenable for graph-theoretical considerations because they are determined by their underlying graph: thetwo-dimensional faces are just the triangles of the graph. In other words, we can think of a Whitney triangulation as anobject wearing two hats: on one hand it is just a graph, and on the other hand it is a 2-dimensional simplicial complexwhich in turn can be considered either as a purely combinatorial object or as a topological surface with a fixed simplicialdecomposition.
It can be shown ([LNLPn02, Prop. 14]) that a simple graph G is the underlying graph of some Whitneytriangulation of a surface if and only if G is locally cyclic. dge-weighted graphs and the energy functionalDefinition 2.10. Let G be an ˜ S -triangulated graph. We say that G is edge-weighted if it is given asystem of edge weights , i.e. a family of positive real numbers ( ω e ) e ∈G ( ) indexed by the set of edges G ( ) , that is invariant under the action of π S .Clearly, the data of a system of edge weights is equivalent to the data of a function η : G ( ) × G ( ) → [ , + ∞) that is symmetric, invariant under the diagonal action of π S , and such that η ( x , y ) > x and y are adjacent. Definition 2.11.
A function η as above is called a pre-kernel on the ˜ S -triangulated graph G .The motivation for introducing this notion will become clear in Definition 2.18 and Proposi-tion 2.19. We are now ready to define the energy functional: Definition 2.12.
Let G be an ˜ S -triangulated graph with a system of edge weights ( ω e ) e ∈G ( ) , and let ρ R : π S → Isom ( N ) be a group homomorphism where N is a metric space. The energy functional E G : Map eq (G , N ) → R is defined by E G ( f ) = (cid:213) e = e xy ∈E ω xy d ( f ( x ) , f ( y )) (12)where E ⊂ G ( ) is any fundamental domain for the action of π S .When N is a Hadamard manifold , E G is a smooth function on the manifold Map eq (G , N ) . Ofcourse we now call a map f ∈ Map eq (G , N ) harmonic when it is a critical point of the energyfunctional E G . When N is not a Hadamard manifold but merely a metric space, one can still define(locally) energy-minimizing harmonic maps.Note that, taking ω e = e ∈ G ( ) and N = R , a harmonic map from G to R in the senseabove coincides with the classical notion of harmonicity for real-valued functions on graphs. Thewell-known mean value property of harmonic functions is generalized: Proposition 2.13.
Let G be an edge-weighted triangulated graph and let N be a metric space.If f ∈ Map eq (G , N ) is an energy-minimizing harmonic map then f ( x ) is a center of mass of theweighted system of points {( f ( y ) , ω xy )} y ∼ x in N for every x ∈ G ( ) . Refer to § 5 for the definition and elementary properties of centers of mass.
Proof. If f ( x ) was not the center of mass of its neighbors, then the part of (12) that involves x couldbe decreased by replacing f ( x ) by the center of mass while leaving the other values unchanged. (cid:3) Vertex weighted-graphs and the Riemannian structureDefinition 2.14.
Let G be an ˜ S -triangulated graph. We say that G is vertex-weighted if it is givena system of vertex weights , i.e. a family of positive real numbers ( µ v ) v ∈G ( ) indexed by the set ofvertices G ( ) , that is invariant under the action of π S . A Hadamard manifold is a complete, simply connected Riemannian manifold of nonpositive curvature. On aHadamard manifold the distance squared function to a fixed point is smooth, while in general it may not be differentiableon the cut locus.
14e think of a system of vertex weights as a π S -invariant Radon measure µ on G ( ) . Of course,this is simply a π S -invariant function µ : G ( ) → ( , + ∞) , but our viewpoint for discretization is toapproximate the smooth theory where µ is the volume density of a Riemannian manifold.Assume now that N is a finite-dimensional Riemannian manifold and let ρ R : π S → Isom ( N ) be a group homomorphism. We saw (Proposition 2.9) that Map eq (G , N ) is a smooth manifold.Moreover, it is easy to describe its tangent space. Proposition 2.15.
The tangent space at f ∈ Map eq (G , N ) is: T f Map eq (G , N ) = Γ eq ( f ∗ T N ) (13) where f ∗ T N is the pullback of the tangent bundle T N to G ( ) and Γ eq ( f ∗ T N ) is its space of π S -equivariant smooth sections . Equivalently, if V ⊆ G ( ) is any fundamental domain for the actionof π S , T f Map eq (G , N ) = (cid:202) x ∈V T f ( x ) N . (14)Notice of course the similarity of (13) with (4). Using the measure µ , one can define a naturalL Riemannian metric on Map eq (G , N ) analogous to (3): Definition 2.16.
Let (G , µ ) be an ˜ S -triangulated vertex-weighted graph, and let ρ R : π S → Isom ( N ) where N is a Riemannian manifold. The L Riemannian metric on Map eq (G , N ) is given by: (cid:104) V , W (cid:105) = ∫ V (cid:104) V x , W x (cid:105) d µ ( x ) where V , W ∈ Γ eq ( f ∗ T N ) and V ⊆ G ( ) is any fundamental domain for the action of π S .Of course, one can write more concretely: (cid:104) V , W (cid:105) = (cid:213) x ∈V µ ( x )(cid:104) V x , W x (cid:105) . One can easily derive that the unit speed geodesics in Map eq (G , N ) are the one-parameterfamilies of functions ( f t ( x )) x ∈G ( ) given by f t ( x ) = exp ( tV x ) , where V ∈ Γ eq ( f ∗ T N ) is a unit vector,and, provided N is connected, the Riemannian distance in Map eq (G , N ) is simply given by d ( f , g ) = (cid:213) x ∈V µ ( x ) d ( f ( x ) , g ( x )) , (15)where on the right hand-side d is the Riemannian distance in N . Of course notice that (15) is justthe discretization of (7). Biweighted graphsDefinition 2.17.
Let G be an ˜ S -triangulated graph (Definition 2.7). We say that G is biweighted ifit is both edge-weighted (Definition 2.10) and vertex-weighted (Definition 2.14).From the discussion of the previous paragraph, when G is an ˜ S -triangulated biweighted graphand N is a Riemannian manifold with a group homomorphism ρ R : π S → N , the space ofequivariant maps Map eq (G , N ) is a Riemannian manifold and the energy is a continuous function E G : Map eq (G , N ) → R . Moreover E G is smooth when N is Hadamard. In § 3 we show that E G is strongly convex under suitable restrictions on ρ R , with an explicit bound on the modulus of15trong convexity (Theorem 3.20). This implies that there exists a unique equivariant harmonic map G → N that can be computed effectively through gradient descent (§ 4).We pause to point out that our definition of the energy functional and harmonic maps in thissetting coincides with Jost’s theory briefly described in § 1.2 (we refer to [Jos96, Jos97] for details).First we introduce the kernel function associated to a biweighted graph: Definition 2.18.
The kernel function associated to a biweighted graph G is the function η : G ( ) × G ( ) → R ( x , y ) (cid:55)→ η ( x , y ) µ ( x ) µ ( y ) where η is the pre-kernel associated to the underlying edge-weighted graph (cf. Definition 2.11)and µ is the measure on G ( ) giving the vertex weights.The next proposition is trivial but conceptually significant: Proposition 2.19.
The energy functional on
Map eq (G , N ) is given by E G ( f ) = ∬ V η ( x , y ) d ( f ( x ) , f ( y )) d µ ( y ) d µ ( x ) where V ⊆ G ( ) × G ( ) is a fundamental domain for the diagonal action of π S . Proposition 2.19 implies that, choosing η r = η for all r >
0, the Jost energy functional E = lim r → E r (compare with (9)) coincides with the energy functional E G . In particular, ournotion of harmonic maps from graphs is a specialization of Jost’s generalized harmonic maps .Next we observe that the Riemannian structure of Map eq (G , N ) allows us to define the discretetension field as: Definition 2.20.
The tension field of f ∈ Map eq (G , N ) is the vector field along f denoted τ G ( f ) ∈ Γ eq ( f ∗ T N ) given by: τ G ( f ) | x = µ ( x ) (cid:213) y ∼ x ω xy exp − f ( x ) ( f ( y )) where we have denoted ω xy the weight of the edge connecting x and y .We have the discrete version of the first variational formula for the energy (Proposition 1.3): Proposition 2.21.
The tension field is minus the gradient of the energy functional: τ G ( f ) = − grad E G ( f ) for any f ∈ Map eq (G , N ) .Proof. In a Riemannian manifold N , when x ∈ N is chosen such that exp x is a diffeomorphism(any x works when N is Hadamard), the function g : x (cid:55)→ d ( x , x ) is smooth and its gradient isgiven by grad g ( x ) = − exp − x ( x ) . (cid:3) It follows, of course, that an equivariant map
G → N is harmonic if and only if its tension fieldis zero, and we obtain a characterization of discrete harmonic maps: Proposition 2.22.
Let G be an edge-weighted triangulated graph and let N be a Hadamard manifold.Then f ∈ Map eq (G , N ) is a harmonic map if and only if f ( x ) is a center of mass of the weightedsystem of points {( f ( y ) , ω xy )} y ∼ x in N for every x ∈ G ( ) .
16e conclude this section by looping back to § 2.1 and the approximation problem. The pointis that when S is equipped with a hyperbolic structure (or more generally any nonpositively curvedmetric), a mesh in the sense of Definition 2.2 induces a biweighted graph structure: Definition 2.23.
Let ρ L : π S → Isom + ( H ) be a Fuchsian representation and let M be an invariantmesh (cf. Definition 2.2). The biweighted graph underlying M is the biweighted graph G such that:• G is the abstract graph underlying M (which is evidently ˜ S -triangulated).• The edge weights are the ω e as in Definition 2.6.• The vertex weights are given by, for every vertex x : µ ( x ) = (cid:213) T Area ( T ) where the sum is taken over all triangles incident to the vertex x .Clearly, any discrete equivariant map along M from H to a Riemannian manifold N inducesan equivariant map G → N , and the energy E M agrees with the energy E G . Of course, the systemsof weights are chosen so that the discrete energy functional E G approximates the smooth energyfunctional (11), and the Riemannian structure of Map eq (G , N ) approximates the L Riemannianmetric on C ∞ ( M , N ) (or L ( M , N ) ), with finer approximation when one takes finer meshes. Theanalysis of this phenomenon is treated in [GLM19]. Remark . With this construction in mind, biweighted triangulated graphs can be roughly thoughtof as follows: the edge weights are a discrete record of the conformal structure of S , and the vertexweights, the area form. Note that the metric structure can be recovered from both, a phenomenonspecific to dimension 2. This is reminiscent of [BPS15]– though distinct– in which two graphs withedge weights are considered conformally equivalent if there is a function of the vertices that scalesone set of weights to another. In this section we study the convexity of the discrete energy functional E G : Map eq (G , H ) → R introduced in the previous section (Definition 2.12). In § 3.1 we recall basics about convexityand strong convexity in Riemannian manifolds. In § 3.2 we review the convexity of the energyfunctional for nonpositively curved target spaces. Next we turn to proving the strong convexity ofthe discrete energy when the target space is H with a Fuchsian representation: we first performsome preliminary computations in the hyperbolic plane in § 3.3, and then prove the main theoremin § 3.4. In § 3.5 we extend this result to Hadamard spaces with negative curvature. The classical notion of convexity in Euclidean vector spaces naturally extends to the Riemanniansetting—as Udrişte puts it [Udr94, Chapter 1],
Riemannian geometry is the natural frame forconvexity .We first give a definition for metric spaces. Recall that geodesics in a metric space ( M , d ) areharmonic maps from intervals of the real line; more concretely, a curve γ : I ⊆ R → M in ( M , d ) is a geodesic if and only if d ( γ ( t ) , γ ( t )) = v | t − t | for any sufficiently close t , t ∈ I , where v is a positive constant. A real-valued function on M is then called (geodesically) convex when it isconvex along geodesics. More precisely: 17 efinition 3.1. Let ( M , d ) be a metric space. A function f : M → R is convex if, for every geodesic γ : [ a , b ] → M and for all t ∈ [ , ] : f ( γ (( − t ) a + tb )) (cid:54) ( − t ) f ( γ ( a )) + t f ( γ ( b )) . When the inequality is strict for all t ∈ ( a , b ) , f is called strictly convex . Furthermore f is called α -strongly convex , where α >
0, if: f ( γ (( − t ) a + tb )) (cid:54) ( − t ) f ( γ ( a )) + t f ( γ ( b )) − α t ( − t ) l ( γ ) where l ( γ ) is the length of γ . The largest such α is called the modulus of strong convexity of f .When M = ( M , g ) is a Riemannian manifold and f is C , one can quickly characterize convexfunctions in terms of the positivity of their Hessian as a quadratic form. Recall that the Hessian of a C function f : M → R is the symmetric 2-covariant tensor field on M defined by Hess ( f ) = ∇( d f ) . Proposition 3.2.
Let f : M → R be a C function on a Riemannian manifold ( M , g ) . Then:• f is convex if and only if it has positive semidefinite Hessian everywhere.• f is strictly convex if it has positive definite Hessian everywhere.• f is α -strongly convex if and only if it has α -coercive Hessian everywhere: ∀ v ∈ T M Hess ( f )( v , v ) (cid:62) α (cid:107) v (cid:107) Convex functions enjoy several attractive properties. Among them, we highlight the straight-forward fact that any sublevel set of a convex function is totally convex (i.e. it contains any geodesicwhose endpoints belong to it). Definition 3.1 and Proposition 3.2 work when M is an infinite-dimensional Riemannian manifold (e.g. C ∞ ( M , N ) as in § 1.1), however note that a convex functionis not necessarily continuous in that case, whereas it is always locally Lipschitz in finite dimension.We refer to [Udr94, Chap. 3] for convex functions on finite-dimensional Riemannian manifolds. We review the convexity of the energy functional when the target is nonpositively curved, whetherin the Riemannian sense or in the sense of Alexandrov, and we also address the possibility of strictor strong convexity in these settings.
Remark . While strict convexity of the energy is a clear-cut way to prove uniqueness of harmonicmaps and strong convexity their existence, neither are necessary. The existence and uniqueness ofharmonic maps has been properly characterized both in the smooth case and in more general spaces:see § 1.3 and § 1.4.
Convexity of the energy in the smooth setting
The second variation of the energy functional in the smooth context was first calculated by Eells-Sampson [ES64] (cf. Proposition 1.4). The next proposition follows immediately from (6):
Proposition 3.4.
Let M be and N be smooth Riemannian manifolds. If N has nonpositive sectionalcurvature, then the Hessian of the energy functional satisfies: ∀ V ∈ Γ ( f ∗ T N ) Hess ( E ) | f ( V , V ) (cid:62) ∫ M (cid:107)∇ V (cid:107) d v g . (16)18ecall that the Hessian of the energy is taken with respect to the L Riemannian structure onthe infinite dimensional manifold C ∞ ( M , N ) .In particular, (16) makes it clear that the energy functional is convex. It is tempting to try andget more out of (16): is E strictly convex? Is it strongly convex? Neither can be true withoutsome obvious restrictions: if f maps into a flat (a totally geodesic submanifold of zero sectionalcurvature), then the energy is constant along the path that consists in translating f along someconstant vector field on the flat. Even when N has negative sectional curvature, whence it has noflats of dimension >
1, this issue remains for constant maps and maps into a curve.However, one can restrict to a connected component of C ∞ ( M , N ) that does not contain suchmaps, and there the question becomes interesting. For example when M is compact and dim N = dim M >
1, the degree of maps is an invariant on the components of C ∞ ( M , N ) , and any componentof nonzero degree contains only surjective map. When the target is negatively curved, (6) doesguarantee strict convexity: Proposition 3.5.
Let M be a Riemannian manifold, let N be a Riemannian manifold of negativesectional curvature. Then the energy functional is strictly convex on any connected component of C ∞ ( M , N ) that does not contain any map of rank everywhere (cid:54) .Proof. Let ( E i ) be a local orthonormal frame in M . The integrand for the Hessian of the energyfunctional (6) is: (cid:107)∇ V (cid:107) − n (cid:213) i = (cid:10) R N ( V , d f ( E i )) d f ( E i ) , V (cid:11) Each term (cid:10) R N ( V , d f ( E i )) d f ( E i ) , V (cid:11) is nonpositive, and is nonzero unless V and d f ( E i ) arecollinear. Indeed, when V and d f ( E i ) are not collinear: (cid:10) R N ( V , d f ( E i )) d f ( E i ) , V (cid:11) = K N ( V , d f ( E i )) (cid:16) (cid:107) V (cid:107) (cid:107) d f ( E i )(cid:107) − (cid:104) V , d f ( E i )(cid:105) (cid:17) < K N ( V , d f ( E i )) denotes the sectional curvature of the plane spanned by V and d f ( E i ) .If Hess ( E ) | f ( V , V ) vanishes, then the integrand must vanish everywhere, so that (1) ∇ V = f ( E i ) and V must be collinear for every i . From (1) it follows that V hasconstant length, and from (2) and the fact that V x (cid:44) x f maps into span ( V x ) forevery x ∈ M . In particular, f has rank (cid:54) (cid:3) As far as the authors are aware, no sufficient conditions for strong convexity of the energyfunctional are known in the smooth setting. We believe that a quantitative refinement of theprevious proof combined with a Poincaré-type inequality should guarantee:
Conjecture 3.6.
Strong convexity holds in the setting of Proposition 3.5.
Convexity of the energy for more general spaces
Defining the energy à la
Jost as in § 1.2, it is straightforward that the energy functional is convexwhen the target space is negatively curved in a suitable sense. Indeed, let ( M , µ ) be a measure spaceand let ( N , d ) be a Hadamard metric space , i.e. a complete CAT ( ) metric space. Recall that aCAT ( ) space is a geodesic metric space where any geodesic triangle T is ‘thinner’ than the triangle T (cid:48) with same side lengths in the Euclidean plane—more precisely, the comparison map T → T (cid:48) isdistance nonincreasing. In a Hadamard space the distance squared function d : N × N → R
19s convex (see [BH99] for details). It follows easily that for any choice of nonnegative symmetrickernel η r (cf § 1.2), the energy functional E r is convex on L ( M , N ) . Furthermore if the energyfunctional E on L ( M , N ) is obtained as a Γ -limit of E r , then it must also be convex [DM93, Thm11.1]. In particular, this applies to our energy functional E G by way of Proposition 2.19: Proposition 3.7.
Let G be any ˜ S -triangulated biweighted graph (Definition 2.17) and let N be aHadamard metric space. The energy functional E G : Map eq (G , N ) → R (Definition 2.12) is convex. We stress that the convexity is relative to a metric structure on Map eq (G , N ) which depends ona system of vertex weights (see Definition 2.14), but the fact that the energy is convex (respectivelystrictly or strongly convex) does not depend on the choice of such vertex weights.We will examine conditions that ensure E G is strongly convex, first for N = H (Theorem 3.18),then in Hadamard manifolds with negative curvature (Theorem 3.20).We highlight some important context: Korevaar-Schoen obtained yet another form of convexityof the energy when the domain M is Riemannian. Their energy functional E , which coincides withJost’s for suitable choices [Chi07], satisfies the convexity inequality E ( f t ) (cid:54) ( − t ) E ( f ) + tE ( f ) − t ( − t ) ∫ M (cid:107)∇ d ( f , f )(cid:107) , (17)where ( f t ) ∈ L ( M , N ) is a geodesic, i.e. f t ( x ) is a geodesic in N for all x ∈ M . This is a weakeranalog of Proposition 3.4. It is again tempting to investigate strong convexity when N has negativecurvature bounded away from 0 and f does not have rank everywhere (cid:54)
1, but work remains to bedone.Mese [Mes02] claims without proof both an improvement of the convexity statement (17) andthe strict convexity of the energy functional at maps of rank (cid:54) ∂ M = ∅ .)Although we prove strong convexity for biweighted graph domains under appropriate restric-tions, we suspect that a more general version of this theorem is true, namely an analog of Conjec-ture 3.6 for singular spaces. In fact, one can further explore extensions to the equivariant setting,with a suitable condition on the target representation strengthening reductivity. In order to study the second variation of the discrete energy for H -valued equivariant maps, wefirst need some convexity estimates in the hyperbolic plane. The strategy in § 3.4 will be to reacha contradiction under the assumption that the second variation of the energy is too small; here wederive necessary consequences of a small second variation of the energy in the the elementary casesconsisting of two and three vertices. We start with a formula for quadrilaterals in H . Proposition 3.8.
Let A , B , C , D be four points in the hyperbolic plane. Let α and β denote theoriented angles as shown in Figure 2. Then: cosh ( DC ) = cosh ( AB ) (cid:2) cosh ( D A ) cosh ( BC ) + sinh ( D A ) sinh ( BC ) cos α cos β (cid:3) − sinh ( AB ) (cid:2) cosh ( D A ) sinh ( BC ) cos β + sinh ( D A ) cosh ( BC ) cos α (cid:3) − sinh ( D A ) sinh ( BC ) sin α sin β . • B • C • D • α β Figure 2 A • B • C • D • α α β Figure 3
Remark . This equation holds without restriction on α and β ; they may be negative or obtuse. Proof.
Referring to Figure 3, the hyperbolic law of cosines implies:cosh ( DC ) = cosh ( D A ) cosh ( AC ) − sinh ( D A ) sinh ( AC ) cos ( α ) . (18)The hyperbolic laws of sines and cosines in the triangle ABC givecosh ( AC ) = cosh ( AB ) cosh ( BC ) − sinh ( AB ) sinh ( BC ) cos ( β ) , and (19)sinh ( AC ) cos ( α ) = sinh ( AC ) cos ( α − α ) = sinh ( AC ) cos ( α ) cos ( α ) + sinh ( AC ) sin ( α ) sin ( α ) = sinh ( AC ) cos ( α ) cos ( α ) + sinh ( BC ) sin ( β ) sin ( α ) . (20)Moreover, it is a consequence of the two forms of the hyperbolic law of cosines (see e.g. [Rat06,p. 82]) in the triangle ABC that we havesinh ( AC ) cos ( α ) = sinh ( AB ) cosh ( BC ) − sinh ( BC ) cosh ( AB ) cos ( β ) . (21)Equation (21) allows us to rewrite equation (20) as:sinh ( AC ) cos ( α ) = (cid:0) sinh ( AB ) cosh ( BC ) − sinh ( BC ) cosh ( AB ) cos ( β ) (cid:1) cos ( α ) + sinh ( BC ) sin ( β ) sin ( α ) . (22)Together (19) and (22) and (18) imply the desired equation. (cid:3) Next we study the convexity of the energy for two points, which amounts to analyzing the secondvariation of the half-distance squared function d : H × H → R . We perform this computation intwo stages: first we study instead the function ( cosh d ) − H × H → R , as it is better suited tocomputations, and then we relate the second variation of the two functions. Proposition 3.10.
Let A and B be two points in the hyperbolic plane at distance D . Let (cid:174) u and (cid:174) v be tangent vectors at A and B respectively. Let A t = exp A ( t (cid:174) u ) and B t = exp B ( t (cid:174) v ) for t ∈ R , andconsider the function F AB ( t ) = cosh ( d ( A t , B t )) − . Then: dd t | t = F AB ( t ) = − sinh ( D ) (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) d d t | t = F AB ( t ) = cosh ( D ) (cid:0) (cid:107) (cid:174) u (cid:107) + (cid:107)(cid:174) v (cid:107) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) cos α cos β (cid:1) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) sin α sin β . where α (resp. β ) is the oriented angle between the geodesic AB and the vector (cid:174) u (resp. (cid:174) v ). roof. Consider the quadrilateral given by the four points A , B , C = B t , D = A t . Note that the angle β here corresponds to the angle π − β of Proposition 3.8. By direct application of Proposition 3.8,1 + F AB ( t ) = cosh ( D ) (cid:2) cosh ( t (cid:107) (cid:174) u (cid:107)) cosh ( t (cid:107)(cid:174) v (cid:107)) − sinh ( t (cid:107) (cid:174) u (cid:107)) sinh ( t (cid:107)(cid:174) v (cid:107)) cos α cos β (cid:3) − sinh ( D ) (cid:2) − cosh ( t (cid:107) (cid:174) u (cid:107)) sinh ( t (cid:107)(cid:174) v (cid:107)) cos β + sinh ( t (cid:107) (cid:174) u (cid:107)) cosh ( t (cid:107)(cid:174) v (cid:107)) cos α (cid:3) − sinh ( t (cid:107) (cid:174) u (cid:107)) sinh ( t (cid:107)(cid:174) v (cid:107)) sin α sin β . The result follows immediately by taking the first and second derivatives at t = (cid:3) Proposition 3.11.
We keep the same setup as Proposition 3.10, and let E AB ( t ) = d ( A t , B t ) .Then: d d t | t = E AB ( t ) = a + b D tanh ( D / ) + c ( D coth D − D tanh ( D / )) , where a , b , and c (cid:62) are given by a = (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) , b = (cid:107) (cid:174) u (cid:107) sin α + (cid:107)(cid:174) v (cid:107) sin β , and c = (cid:0) (cid:107) (cid:174) u (cid:107) sin α − (cid:107)(cid:174) v (cid:107) sin β (cid:1) . Proof.
For ease in notation, we leave the subscripts AB from E AB and F AB in what follows. Wehave E ( t ) = φ ◦ F ( t ) , where φ ( x ) = ( arcosh ( + x )) . It is straightforward to check that φ (cid:48) ( cosh x − ) = x sinh x , and φ (cid:48)(cid:48) ( cosh x − ) = sinh x − x cosh x sinh x . Since we have E (cid:48)(cid:48) ( ) = φ (cid:48)(cid:48) ( F ( )) ( F (cid:48) ( )) + φ (cid:48) ( F ( )) F (cid:48)(cid:48) ( ) , using Proposition 3.10 we find that E (cid:48)(cid:48) ( ) = sinh D − D cosh D sinh D · sinh D (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) + D sinh D (cid:16) cosh D (cid:16) (cid:107) (cid:174) u (cid:107) + (cid:107)(cid:174) v (cid:107) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) cos α cos β (cid:17) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) sin α sin β (cid:17) = (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) + D coth D (cid:16) − (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) + (cid:16) (cid:107) (cid:174) u (cid:107) + (cid:107)(cid:174) v (cid:107) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) cos α cos β (cid:17) (cid:17) − D csch D (cid:0) (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) sin α sin β (cid:1) = a + b D coth D + ( c − b ) D csch D . To finish, note that coth D − csch D = tanh ( D / ) . (cid:3) The following quantitative control is at the core of strong convexity for E G : Proposition 3.12.
Let E AB ( t ) be the function as in Proposition 3.11. We have d d t | t = E AB ( t ) (cid:62) (cid:107) (cid:174) u − P [ BA ] (cid:174) v (cid:107) , where P [ BA ] (cid:174) v denotes the parallel transport of (cid:174) v along the geodesic segment B A . roof. By rewriting E (cid:48)(cid:48) AB ( ) using Proposition 3.11, and noting that 2 b (cid:62) c by the Cauchy-Schwarzinequality, we find E (cid:48)(cid:48) AB ( ) = a + b D tanh D + c (cid:18) D coth D − D tanh D (cid:19) (cid:62) a + c (cid:18) D coth D − D D (cid:19) = a + c · D D . Because x coth x (cid:62)
1, we find that E (cid:48)(cid:48) AB ( ) (cid:62) a + c . The proof is completed by checking that thequantity a + c is precisely (cid:107) (cid:174) u − P [ BA ] (cid:174) v (cid:107) : a + c = (cid:0) (cid:107) (cid:174) u (cid:107) cos α − (cid:107)(cid:174) v (cid:107) cos β (cid:1) + (cid:0) (cid:107) (cid:174) u (cid:107) sin α − (cid:107)(cid:174) v (cid:107) sin β (cid:1) = (cid:107) (cid:174) u (cid:107) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) cos ( α − β ) + (cid:107)(cid:174) v (cid:107) = (cid:104)(cid:174) u , (cid:174) u (cid:105) − (cid:104)(cid:174) u , P [ BA ] (cid:174) v (cid:105) + (cid:104)(cid:174) v , (cid:174) v (cid:105) = (cid:107) (cid:174) u − P [ BA ] (cid:174) v (cid:107) . (cid:3) H Let G be any ˜ S -triangulated biweighted graph (Definition 2.17) and let ρ R : π S → Isom ( H ) be aFuchsian representation. We are ready to prove strong convexity of the discrete energy functional E G : Map eq (G , H ) → R introduced in Definition 2.12.Choose once and for all a fundamental domain for the action of π S on G ( ) , consisting ofvertices { p , . . . , p n } ⊆ G ( ) . Recall that G is equipped with vertex and edge weights; these arecompletely determined by the weights µ i = µ ( p i ) and ω ij = ω ( e p i p j ) for i , j ∈ { , . . . , n } .Fix an equivariant map f ∈ Map eq (G , H ) , recorded by the tuple ( x , . . . , x n ) ∈ ( H ) n where x i (cid:66) f ( p i ) . Also fix a tangent vector (cid:174) v ∈ T f Map eq (G , H ) , given by (cid:174) v = ((cid:174) v , . . . , (cid:174) v n ) where v i ∈ T x i H as in (14). We assume that (cid:174) v is a unit tangent vector: by Definition 2.16 this means (cid:205) i µ i (cid:107) (cid:174) v i (cid:107) = t = E G ( t ) (cid:66) E G ◦ exp f ( t (cid:174) v ) . Let us denote E ij ( t ) (cid:66) d ( exp x i ( t (cid:174) v i ) , exp x j ( t (cid:174) v j )) for each edge e ij between points p i and p j .First we observe that if the second variation of E G is small, then Proposition 3.12 implies thatthe tangent vectors (cid:174) v i all have approximately the same length. Precisely, Lemma 3.13.
For all i we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) (cid:174) v i (cid:107) − √ A (cid:12)(cid:12)(cid:12)(cid:12) < (cid:114) E (cid:48)(cid:48)G ( ) D Ω , where D is the diameter of the quotient graph G/ π S , A = (cid:205) i µ i , and Ω = min ω ij .Proof. Let ε ij = E (cid:48)(cid:48) ij ( ) for each edge e ij ∈ E , so that E (cid:48)(cid:48)G ( ) = (cid:205) e i j ∈E ω ij ε ij . Observe that foreach edge e ij ∈ E , one finds that |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107)| < √ ε ij by Proposition 3.12. For any pair of points p i and p j , choose a path p i = p i , p i , . . . , p i r = p j and observe that by the triangle inequality andthe Cauchy-Schwarz inequality we have: (cid:12)(cid:12) (cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107) (cid:12)(cid:12) (cid:54) r (cid:213) k = √ ε i k − i k (cid:54) (cid:32) r (cid:213) k = ε i k − i k (cid:33) / · (cid:32) r (cid:213) k = (cid:33) / . r (cid:54) D , and Ω (cid:205) ε ij (cid:54) E (cid:48)(cid:48)G ( ) , so the above inequality implies (cid:12)(cid:12) (cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107) (cid:12)(cid:12) < (cid:115) E (cid:48)(cid:48)G ( ) D Ω . Let δ = (cid:113) E (cid:48)(cid:48)G ( ) D / Ω . As |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107)| < δ for all i , j , |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107) | = |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107)| · |(cid:107) (cid:174) v i (cid:107) + (cid:107) (cid:174) v j (cid:107)|≤ |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107)| · (|(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107)| + (cid:107) (cid:174) v i (cid:107)) < δ + δ (cid:107) (cid:174) v i (cid:107) . As (cid:174) v is a unit tangent vector and A = (cid:205) j µ j , we have1 = (cid:213) j µ j (cid:107) (cid:174) v j (cid:107) = A (cid:107) (cid:174) v i (cid:107) + (cid:213) j µ j ((cid:107) (cid:174) v j (cid:107) − (cid:107) (cid:174) v i (cid:107) ) . Rearranging and taking absolute values we find | A (cid:107) (cid:174) v i (cid:107) − | (cid:54) (cid:213) j µ j |(cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107) | < A ( δ + δ (cid:107) (cid:174) v i (cid:107)) . Dividing by A we obtain (cid:12)(cid:12)(cid:12)(cid:12) (cid:107) (cid:174) v i (cid:107) − A (cid:12)(cid:12)(cid:12)(cid:12) < δ + δ (cid:107) (cid:174) v i (cid:107) . This implies that we have (cid:107) (cid:174) v i (cid:107) − δ (cid:107) (cid:174) v i (cid:107) − δ < A , and (cid:107) (cid:174) v i (cid:107) + δ (cid:107) (cid:174) v i (cid:107) + δ > A . In other words, ((cid:107) (cid:174) v i (cid:107) − δ ) < A + δ , and ((cid:107) (cid:174) v i (cid:107) + δ ) < A . We conclude that √ A − δ < (cid:107) (cid:174) v i (cid:107) < δ + (cid:113) A + δ < √ A + ( + √ ) δ < √ A + δ , as desired. (cid:3) Together with Proposition 3.12, this is enough to produce a lower bound for the second variation:
Proposition 3.14.
We have d d t | t = E G ( t ) (cid:62) A (cid:18) + (cid:113) D Ω (cid:19) , (23) where D is the diameter of the quotient graph G/ π S , A = (cid:205) i µ i , and Ω = min ω ij is the minimumedge weight.Remark . Observe that this demonstrates that there is a constant C > G sothat the modulus of convexity of E G is at least C Ω AD . When G is the biweighted graph underlying aFuchsian representation as in Definition 2.23, the modulus of convexity is at least C Ω | χ ( S )| D .24 roof. Suppose towards contradiction that E (cid:48)(cid:48)G ( ) < A − (cid:18) + (cid:113) D Ω (cid:19) − , i.e. (cid:113) E (cid:48)(cid:48)G ( ) < √ A − (cid:114) E (cid:48)(cid:48)G ( ) · D Ω . (24)Rearranging, we find that (cid:113) E (cid:48)(cid:48)G ( ) √ A − (cid:113) E (cid:48)(cid:48)G ( ) · D Ω < . Now consider an edge x i ∼ x j in G , let δ ij = (cid:107) (cid:174) v i − P [ x j x i ] (cid:174) v j (cid:107) , and let θ ij indicate the principalvalue of the angle formed between (cid:174) v i and P [ x j x i ] (cid:174) v j . Evidently, (cid:107) (cid:174) v i (cid:107)| θ ij | is bounded by π times thedistance between (cid:174) v i and (cid:107) (cid:174) v i (cid:107)(cid:107) (cid:174) v j (cid:107) P [ x j x i ] (cid:174) v j , and the latter distance is bounded by δ ij + (cid:12)(cid:12) (cid:107) (cid:174) v i (cid:107) − (cid:107) (cid:174) v j (cid:107) (cid:12)(cid:12) (cid:54) δ ij .Therefore we have (cid:107) (cid:174) v i (cid:107)| θ ij | (cid:54) π · δ ij . It is not hard to see that (24) implies 3 ( E (cid:48)(cid:48)G ( ) · D / Ω ) / < √ A , so by Lemma 3.13 we conclude that (cid:107) (cid:174) v i (cid:107) (cid:62) √ A − ( E (cid:48)(cid:48)G ( ) · D / Ω ) / >
0. In that case we may rearrange the above to find that | θ ij | (cid:54) π δ ij (cid:107) (cid:174) v i (cid:107) (cid:54) π · (cid:113) E (cid:48)(cid:48)G ( ) √ A − (cid:113) E (cid:48)(cid:48)G ( ) · D / Ω < π , (25)by our assumption (24).Now extend the vectors (cid:174) v i at the vertices of G over each of the edges of the graph G ⊂ Y byinterpolating linearly. Consider a triangle T in G with vertices x i , x j , x k and boundary contour γ ,oriented counterclockwise. The winding number of (cid:174) V with respect to γ , denoted ω γ ( (cid:174) V ) , is given by ω γ ( (cid:174) V ) = π (cid:0) θ ij + θ jk + θ ki + θ i + θ j + θ k (cid:1) , where θ i , θ j , θ k are the exterior angles of T at x i , x j , x k , respectively. Evidently, the sum of theexterior angles is at least π , and at most 3 π . Together with (25), we find that 0 < ω γ ( (cid:174) V ) < ω γ ( (cid:174) V ) = T . Now (cid:174) V can be extended over the triangles to anonvanishing section of f ∗ T Y , contradicting Lemma 3.16. (cid:3) Lemma 3.16.
Suppose that f : S → Y is a map between surfaces with χ ( Y ) (cid:44) and deg f (cid:44) .Then f ∗ T Y has no nonvanishing sections.Proof. Let e indicate the Euler class . We have (cid:104) e ( f ∗ T Y ) , [ S ](cid:105) = (cid:104) f ∗ e ( T Y ) , [ S ](cid:105) = (cid:104) e ( T Y ) , f ∗ [ S ](cid:105) = deg f · χ ( Y ) , which is nonzero by assumption. Therefore e ( f ∗ T Y ) (cid:44)
0, and f ∗ T Y has no nonvanishing sections. (cid:3) The discrete heat flow will also require an upper bound for the Hessian of E G (see § 4).25 roposition 3.17. Suppose that E G ( ) (cid:54) E . Then d d t | t = E G ( t ) (cid:54) VWU (cid:32) + (cid:114) E Ω coth (cid:114) E Ω (cid:33) (cid:67) β (26) where V is the maximum valence of vertices of G , U = min { µ i } is the minimum vertex weight, Ω = min { ω ij } is the minimum edge weight, and W = max { ω ij } is the maximum edge weight.Proof. It is not hard to see from Proposition 3.11 that we haved d t | t = E ij ( t ) (cid:54) (cid:16) (cid:107) (cid:174) v i (cid:107) + (cid:107) (cid:174) v j (cid:107) (cid:17) (cid:0) + d ( x i , x j ) coth d ( x i , x j ) (cid:1) . Letting L = max { d ( x i , x j )} we findd d t | t = E G ( t ) = (cid:213) e i j ∈E ω ij d d t | t = E ij ( t ) (cid:54) W ( + L coth L ) (cid:213) e i j ∈E (cid:16) (cid:107) (cid:174) v i (cid:107) + (cid:107) (cid:174) v j (cid:107) (cid:17) (cid:54) WU ( + L coth L ) (cid:213) e i j ∈E (cid:16) µ i (cid:107) (cid:174) v i (cid:107) + µ j (cid:107) (cid:174) v j (cid:107) (cid:17) (cid:54) VWU ( + L coth L ) (cid:213) i µ i (cid:107) (cid:174) v i (cid:107) = VWU ( + L coth L ) . The assumption E G ( ) (cid:54) E implies that Ω L (cid:54) E , so we are done. (cid:3) Together, Proposition 3.17 and Proposition 3.14 imply:
Theorem 3.18.
Suppose that G is a biweighted triangulated graph, N = H is the hyperbolic planeand ρ = ρ R : π S → Isom + ( H ) is Fuchsian. Then the energy functional E G : Map eq (G , N ) → R is strongly convex. More precisely, ∀ (cid:174) v α (cid:107)(cid:174) v (cid:107) (cid:54) Hess ( E G )((cid:174) v , (cid:174) v ) where α is given explicitly by (23) . Moreover, on the compact convex set { E (cid:54) E } ⊆ Map eq (G , N ) , ∀ (cid:174) v Hess ( E G )((cid:174) v , (cid:174) v ) (cid:54) β (cid:107)(cid:174) v (cid:107) where β is given explicitly by (26) . Some of the steps in the proof of Theorem 3.18 actually hold in much greater generality. Forinstance, Proposition 3.12 holds verbatim if we replace H with any Hadamard manifold. Keepingthe same setup as in § 3.3, let A , B ∈ N , let (cid:174) u and (cid:174) v be vectors in T A N and T B N , respectively, let A t = exp A ( t (cid:174) u ) and B t = exp B ( t (cid:174) v ) , and let E AB ( t ) = d N ( A t , B t ) . Proposition 3.19.
Suppose that N is a Hadamard manifold. Then we have d d t | t = E AB ( t ) (cid:62) (cid:107) (cid:174) u − P [ BA ] (cid:174) v (cid:107) . The proof below builds on § 3.3. 26 roof.
We start by noting that the computations for E (cid:48)(cid:48) AB ( ) (Proposition 3.10 and Proposition 3.11)simplify dramatically when N = R . In this case we haved d t E AB ( t ) = (cid:107) (cid:174) u − (cid:174) v (cid:107) . (27)In particular, if (cid:174) u (cid:44) (cid:174) v then E AB ( t ) is (cid:107) (cid:174) u − (cid:174) v (cid:107) -strongly convex.Now we turn to the general case. Let r , s ∈ R , and consider the quadrilateral through A r , B r , B s ,and A s . Because Hadamard manifolds are CAT ( ) , this quadrilateral has a comparison quadrilateralin R with vertices A (cid:48) r , B (cid:48) r , B (cid:48) s , A (cid:48) s . For any t ∈ R , as a consequence of (27) we have d ( A ( − t ) r + ts , B ( − t ) r + ts ) (cid:54) d ( A (cid:48)( − t ) r + ts , B (cid:48)( − t ) r + ts ) (cid:54) ( − t ) d ( A (cid:48) r , B (cid:48) r ) + td ( A (cid:48) s , B (cid:48) s ) − (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) t ( − t ) | r − s | (cid:54) ( − t ) d ( A r , B r ) + td ( A s , B s ) − (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) t ( − t ) | r − s | , where (cid:174) u (cid:48) = A (cid:48) s − A (cid:48) r and (cid:174) v (cid:48) = B (cid:48) s − B (cid:48) r in R , respectively. In terms of E AB we have E AB (( − t ) r + ts ) (cid:54) ( − t ) E AB ( r ) + tE AB ( s ) − (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) t ( − t ) | r − s | . This is almost an equivalent formulation of the strong convexity of E AB (see Definition 3.1),however the term (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) depends on both r and s . In fact, this detail is essential, as E AB mayfail to be strongly convex. For δ >
0, taking r = s = δ and t = , one finds E AB ( ) = E AB (cid:18) δ + (− δ ) (cid:19) (cid:54) E AB ( δ ) + E AB (− δ ) − (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) δ . (We stress the dependence of (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) on δ .) Rearranging we find that (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) (cid:54) E AB ( δ ) − E AB ( ) + E AB (− δ ) δ . (28)As δ →
0, the right-hand side approaches d d t | t = E AB ( t ) . As for the left-hand side, we may rewrite (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) = (cid:107) (cid:174) u (cid:48) (cid:107) + (cid:107)(cid:174) v (cid:48) (cid:107) − (cid:107) (cid:174) u (cid:48) (cid:107) (cid:107)(cid:174) v (cid:48) (cid:107) cos ( α (cid:48) − β (cid:48) ) = (cid:107) (cid:174) u δ (cid:107) + (cid:107)(cid:174) v δ (cid:107) − (cid:107) (cid:174) u δ (cid:107) (cid:107)(cid:174) v δ (cid:107) cos ( α (cid:48) − β (cid:48) ) , where (cid:174) u δ = exp − A − δ ( A δ ) and (cid:174) v δ = exp − B − δ ( B δ ) (so that (cid:107) (cid:174) u δ (cid:107) = (cid:107) (cid:174) u (cid:48) (cid:107) and (cid:107)(cid:174) v δ (cid:107) = (cid:107)(cid:174) v (cid:48) (cid:107) ), and α (cid:48) (resp. β (cid:48) ) are the oriented angles (measured in R ) between the geodesic A − δ B − δ and (cid:174) u (cid:48) (resp. (cid:174) v (cid:48) ). Bya well-known theorem of Alexandrov (see [Ale51] or [BH99]), the interior angles of a quadrilateralof N are smaller than those of the model, and we conclude that α δ (cid:54) α (cid:48) and π − β δ (cid:54) π − β (cid:48) , where α δ and β δ are the interior angles at A − δ and B − δ , respectively, of the quadrilateral with vertices A − δ , B − δ , B δ , and A δ . Thus α δ − β δ (cid:54) α (cid:48) − β (cid:48) , and we may conclude that (cid:107) (cid:174) u (cid:48) − (cid:174) v (cid:48) (cid:107) (cid:62) (cid:107) (cid:174) u δ (cid:107) + (cid:107)(cid:174) v δ (cid:107) − (cid:107) (cid:174) u δ (cid:107) (cid:107)(cid:174) v δ (cid:107) cos ( α δ − β δ ) . Using the latter for the lefthand side of (28) and taking the limit as δ →
0, we find that (cid:107) (cid:174) u (cid:107) + (cid:107)(cid:174) v (cid:107) − (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) cos ( α − β ) (cid:54) d d t | t = E AB ( t ) . The lefthand side here is precisely (cid:107) (cid:174) u − P [ BA ] (cid:174) v (cid:107) , as in the proof of Proposition 3.12. (cid:3) Theorem 3.20.
Let G be a biweighted ˜ S -triangulated graph, let N be a two-dimensional closedRiemannian manifold of nonzero Euler characteristic and nonpositive sectional curvature andsuppose that ρ : π S → π N is a group homomorphism induced by a homotopy class of maps S → N of nonzero degree. Then the energy functional E G : Map eq (G , ˜ N ) → R is strongly convexwith modulus of convexity given by (23) .Proof. Proposition 3.19 may take the place of Proposition 3.12, and we see that Lemma 3.13 holdsfor any nonpositively curved target. Now the proof of Proposition 3.14 holds verbatim. (The readercan observe that the fourth sentence of the last paragraph still holds in the setting of nonpositivecurvature: the sum of exterior angles of a geodesic triangle is between π and 3 π .) (cid:3) The area of mathematics concerned with methods for finding the minima of a convex function F : Ω → R , called convex optimization , has been intensely developed in the last few decadesand finds countless applications. The majority of the existing literature deals with the classicalcase where Ω is a subset of a Euclidean space; the Riemannian setting has been far less exploredalthough it is a natural and useful extension. Udrişte’s book [Udr94] is a good standard referencefor Riemannian convex optimization (see e.g. [AMS08], [ZS16] for more recent developments).Our goal is to present a simple but effective method that can be implemented to find the minimumof the discrete energy functional, with a rigorous proof of convergence and explicit control ofthe convergence rate. Of course, there are more advanced and faster algorithms in practice. Forexample, the C++ library ROPTLIB [HAGH16] was developed for this purpose.A gradient descent method is an iterative algorithm for minimizing a function F : Ω ⊆ R N → R which produces a sequence ( x k ) k (cid:62) of points in Ω , defined inductively by: x k + = x k − t k grad F ( x k ) . (29)In this relation, t k (cid:62) stepsize . If F has good convexity properties such as beingstrongly convex, then a small enough stepsize t k = t > ( x k ) to a minimum of F , with explicit control of the convergence rate.This method naturally extends to the Riemannian setting, i.e. when F : Ω ⊆ M → R is definedon a subset Ω of a Riemannian manifold M , in which case (29) should be understood as x k + = exp x k (− t k grad F ( x k )) . (30)As in the Euclidean setting, − grad F ( x k ) is the direction of steepest descent for F at x k , so it isnatural to look for x k + in the geodesic ray based at x k in this direction. Note that the gradientdescent method can simply be described as Euler’s method for the gradient flow ODE: x (cid:48) ( t ) = − grad F ( x ( t )) . radient descent method with fixed stepsize for strongly convex functions The gradient descent method with fixed stepsize remains valid for C strongly convex functions onRiemannian manifolds: Theorem 4.1 ([Udr94, Chap. 7, Theorem 4.2]) . Let ( M , g ) be a complete Riemannian manifoldand let F : M → R be a function of class C . Assume that there exists α, β > such that: ∀ v ∈ T M α (cid:107) v (cid:107) (cid:54) ( Hess F )( v , v ) (cid:54) β (cid:107) v (cid:107) Then F has a unique minimum x ∗ . Furthermore, for t ∈ ( , β ] , the gradient descent method withfixed stepsize t k = t converges to x ∗ with a linear convergence rate: d ( x k , x ∗ ) (cid:54) c q k The constants c (cid:62) and q ∈ [ , ) are given by: c = (cid:114) α ( F ( x ) − F ( x ∗ )) q = (cid:115) − t α (cid:18) + αβ (cid:19) . (31) Remark . A key step in the proof of Theorem 4.1 is that ( F ( x k ) − F ( x ∗ )) is nonincreasing andlimits to 0 with a linear convergence rate. In particular, ( F ( x k )) k (cid:62) is nonincreasing, therefore anysublevel set of F is stable under the gradient descent. Moreover, such a set is convex and compactby strong convexity of F . Thus, the gradient descent method is valid even if the Hessian of is notbounded above: one can restrict to a sublevel set, where the Hessian of F is bounded. Gradient descent method with optimal stepsize for strongly convex functions
There are many variants of the gradient descent method that can be more or less useful dependingon the context (see e.g. [ZS16], [FLP18]). One of them is the optimal stepsize gradient descent , aninstance of the gradient descent method (30) where one performs a line search in order to determinea stepsize t k that minimizes F ( x k + ) . Clearly, when F is strongly convex, such a t k exists, is unique,and is > x k = x ∗ . When the Hessian of F is known analytically, Newton’s method offers avery fast line search. Theorem 4.3.
Let F : M → R as in Theorem 4.1. The optimal stepsize gradient descent has alinear convergence rate at least as fast as that of Theorem 4.1, for any choice of the fixed stepsize.Proof. Theorem 4.3 is derived from a careful analysis of the proof of Theorem 4.1 which can befound in [Udr94, Chapter 7, Theorem 4.2]. This proof is a combination of three observations:(i) For any x ∈ M : α d ( x , x ∗ ) (cid:54) F ( x ) − F ( x ∗ ) (cid:54) β d ( x , x ∗ ) . (32)This follows from a Taylor expansion of F at x ∗ along the geodesic [ x ∗ , x ] .(ii) For any x ∈ M : (cid:107) grad F ( x )(cid:107) (cid:62) α (cid:18) + αβ (cid:19) ( F ( x ) − F ( x ∗ )) . This follows from a Taylor expansion of F at x along the geodesic [ x , x ∗ ] and from (32).29iii) For any x ∈ M and for any t ∈ [ , β ] : F ( x ) − F ( x + ( t )) (cid:62) t (cid:107) grad F ( x )(cid:107) where x + ( t ) (cid:66) exp x (− t grad F ( x )) . This follows from a Taylor formula for F along [ x , x + ( t )] .It follows immediately from these three observations that for any x ∈ M and for any t ∈ [ , β ] : F ( x + ( t )) − F ( x ∗ ) (cid:54) Q ( t ) ( F ( x ) − F ( x ∗ )) (33)where Q ( t ) = − t α ( + αβ ) = q .When one performs the gradient descent method with fixed stepsize t , by assumption x k + = x + k ( t ) . Theorem 4.3 is then easily concluded by finding F ( x k ) − F ( x ∗ ) (cid:54) Q k ( F ( x ) − F ( x ∗ )) from(33) (with an obvious induction) and making one last use of (32).If instead one performs an optimal stepsize gradient descent, then x k + = x + k ( t k ) , where t k is the optimal step. Fix t ∈ [ , β ] . By definition of the optimal step, F ( x + k ( t k )) (cid:54) F ( x + k ( t )) , so F ( x k + ) − F ( x k ) (cid:54) F ( x + k ( t )) − F ( x k ) . Therefore we can derive from (33) that F ( x k + ) − F ( x ∗ ) (cid:54) Q ( t ) ( F ( x k ) − F ( x ∗ )) and the conclusion follows like before. (cid:3) The discrete heat flow can be described as a discretization both in time and space of the heat flow on C ∞ ( M , N ) . Recall that the smooth heat flow is the gradient flow of the smooth energy functional:dd t f t = τ ( f t ) where τ ( f t ) = − grad E ( f t ) is the tension field of f t (cf § 1.1, § 1.3).We recall the setup of our discretization: Let G be a biweighted ˜ S -triangulated graph (Defi-nition 2.17), let N be a Riemannian manifold, and let ρ : π S → Isom ( N ) be a group homomo-morphism. Recall that the discrete energy is a function E G : Map eq (G , N ) → R (Definition 2.12),where Map eq (G , N ) is the space of ρ -equivariant maps G → N . The latter space has a naturalRiemannian structure with respect to which the gradient of the energy is minus the discrete tensionfield τ G (Definition 2.16 and Proposition 2.21). Thus we define the discrete heat flow: Definition 4.4.
The discrete heat flow is the iterative algorithm which, given f ∈ Map eq (G , N ) ,produces the sequence ( f k ) k ∈ N in Map eq (G , N ) defined inductively by the relation f k + ( x ) = exp f k ( x ) (cid:0) t k ( τ G f k ) x (cid:1) , where t k ∈ R is a chosen stepsize.The main theorem of this section is an immediate application of Theorem 3.20 and Theorem 4.1: Theorem 4.5.
Let G be a biweighted ˜ S -triangulated graph. Let N be two-dimensional Hadamardmanifold of nonpositive curvature and ρ : π S → Isom ( N ) is a faithful representation whose imageis contained in a discrete subgroup of Isom ( N ) acting freely and properly. Then there exists aunique ρ -equivariant harmonic map f ∗ : G → N . Moreover, for any f ∈ Map eq (G , N ) and for any . . . . × × × × × (cid:96) = . (cid:96) = . (cid:96) = . (cid:96) = . Harmony . sufficiently small t > , the discrete heat flow with initial value f and fixed stepsize t converges to f ∗ with a linear convergence rate: d ( f k , f ∗ ) (cid:54) cq k (34) where c > and q ∈ [ , ) are constants, and d ( f k , f ∗ ) is the L distance in Map eq (G , N ) . Of course, it also follows from Theorem 4.3 that the discrete heat flow with optimal stepsizeconverges to f ∗ as well, with a linear convergence rate as least as fast as (34).We emphasize that in our favorite setting where N = H and ρ : π S → Isom + ( H ) is Fuchsian,Theorem 3.18 enables explicit estimates on the constants c and q in (34): the expressions of c and q are given by (31), in which α is given by (23) and β is given by (26) with E = E ( f ) . In Figure 4 and Figure 5 we present numerical experiments performed with the software
Harmony . Comparison of different fixed stepsizes
In the first experiment (Figure 4) we observe the number of iterations required for the discrete heatflow with fixed stepsize to converge as a function of the stepsize.Let S be a closed oriented surface of genus 2. We fix a domain Fuchsian representation ρ L : π S → Isom + ( H ) : the representation pictured on the left in Figure 8, and let Harmony construct an invariant mesh (depth 4, 1921 vertices). We let the target Fuchsian representation ρ R vary, taking Fenchel-Nielsen lengths ( , , (cid:96) ) and twists (− . , , . ) , where (cid:96) ∈ { . , . , . , . } .We observe that the plotted points resemble in profile functions of the form − C ( log ( − C t )) − ,which is precisely the type of function predicted by Theorem 4.5.31umber of iterations5 × × × × (cid:96) fixed stepsizeoptimal stepsizecosh-center of massFigure 5: Comparison of the three methods performed by Harmony . Comparison of our three methods
For the second experiment (Figure 5) we compare the convergence rate, in terms of number ofiterations, of our three methods:• Discrete heat flow with fixed stepsize (see § 4.2),• Discrete heat flow with optimal stepsize (see § 4.2),• Cosh-center of mass method (see § 5.3).We keep the same setting as before, letting (cid:96) this time vary between 0 . .
4. As the figureshows, the cosh-center of mass method is more effective than either gradient descent methods.
In this section we investigate a center of mass algorithm towards the minimization of the discreteenergy. We shall see that it is in some sense a variant of the fixed stepsize discrete heat flow. In thecurrent state of our software
Harmony , it is the most effective method (see § 4.3).First we recall facts about centers of mass in Riemannian manifolds and investigate how theyrelate to harmonic maps. We shall prove in particular a generalized mean value property forharmonic maps between Riemannian manifolds (Theorem 5.9).
Let ( Ω , F , µ ) be a probability space, ( X , d ) be a metric space, and h : Ω → X a measurable map. Definition 5.1. A center of mass (or barycenter ) of h is a minimizer of the function P h : X → R x (cid:55)→ ∫ Ω d ( x , h ( y )) d µ ( y ) . (35)32n general, neither existence nor uniqueness of centers of mass hold. If X is a Hadamard spaceand h ∈ L ( Ω , X ) then existence and uniqueness do hold [KS93, Lemma 2.5.1]. For Riemannianmanifolds we have: Theorem 5.2 (Karcher [Kar77]) . Assume that X is a complete Riemannian manifold and h takesvalues in a ball B = B ( x , r ) ⊂ X such that:• B is strongly convex : any two points of B are joined by a unique minimal geodesic γ : [ , ] → X , and each such geodesic maps entirely into B .• B has nonpositive sectional curvature, or r < π √ K where K > is an upper bound for thesectional curvature in B .Under these conditions, the function P h of (35) only has interior minimimizers on ¯ B and is stronglyconvex inside B . Consequently, existence and uniqueness of the center of mass hold. Note that if X is a complete Riemannian manifold, any sufficiently small r > h is the unique point G ∈ X such that ∫ Ω exp − G ( h ( y )) d µ ( y ) = . (36)This simply expresses the vanishing at G of the gradient of the function P h : X → R of (35).Of course, Definition 5.1 generalizes the usual notion of center of mass in R n : when h ∈ L ( Ω , R n ) , the center of mass given by G = ∫ Ω h ( y ) d µ ( y ) . When X is not Euclidean, the center ofmass is only defined implicitly, but one can estimate proximity to the center of mass: Lemma 5.3.
Assume that the conditions of Theorem 5.2 are satisfied. In particular the center ofmass G of h is well-defined. If G (cid:48) is a point in X such that: (cid:13)(cid:13)(cid:13)(cid:13)∫ Ω exp − G (cid:48) ( h ( y )) d µ ( y ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:54) δ then d ( G , G (cid:48) ) (cid:54) C δ , where C = when X has nonpositive curvature, or C = C ( K , r ) > when X has sectional curvaturebounded above by K > in a strongly convex ball of radius r containing the image of h .Proof. This is an immediate consequence of the fact that the function P h of (35) is C -stronglyconvex under the assumptions of the lemma. Also see [Kar77, Thm 1.5]. (cid:3) In this section we show a generalized mean value property for smooth harmonic maps betweenRiemannian manifolds. Let f : ( M , g ) → ( N , h ) be a smooth map between Riemannian manifolds.Fix x ∈ M . Denote by S r (resp. B r ) the sphere (resp. the closed ball) centered at the origin ofradius r in the Euclidean vector space ( T x M , g ) . Also denote ˆ S r (resp. ˆ B r ) the sphere (resp. theclosed ball) centered at x of radius r in ( M , g ) . The topological space S r (resp. B r ) can be equippedwith a natural Borel probability measure by taking the measure induced from the Euclidean metric g in T x M , renormalized so that it has total mass 1. Similarly, ˆ S r (resp. ˆ B r ) can be equipped with anatural Borel probability measure by taking the measure induced from the Riemannian metric g . Definition 5.4.
We define four functions S r f , B r f , ˆ S r f , ˆ B r f : M → N as follows. Given x ∈ M :• S r f ( x ) (resp. B r f ( x ) ) is the center of mass of f ◦ exp x : S r → N (resp. f ◦ exp x : B r → N ).33 ˆ S r f ( x ) (resp. ˆ B r f ( x ) ) is the center of mass of f | ˆ S r : ˆ S r → N (resp. f | ˆ B r : ˆ B r → N ). Remark . The four functions of Definition 5.4 are well-defined as long as ( N , h ) is a Hadamardmanifold, or as long as r is small enough and ( N , h ) has sectional curvature bounded above andinjectivity radius bounded below by a positive number (e.g. N is compact).Note that S r f ( x ) and ˆ S r f ( x ) are different in general, as are B r f ( x ) and ˆ B r f ( x ) . However theyare very close when r is small: Proposition 5.6.
Let f : M → N be a smooth map. Then for all x ∈ M : d ( S r f ( x ) , ˆ S r f ( x )) = O ( r ) d ( B r f ( x ) , ˆ B r f ( x )) = O ( r ) . Proof.
The proof is technical but not very difficult. It is basically derived from a Taylor expansionof the metric in normal coordinates at x and one use of Lemma 5.3. We will do several similarproofs in what follows, so we skip the details for brevity. (cid:3) Of course, in the case where M = R m and N = R (or N = R n ), S r f and ˆ S r f (resp. B r f andˆ B r f ) coincide. We recall the celebrated mean property for harmonic functions in this setting: Theorem 5.7. f : R m → R is harmonic if and only if S r f = B r f = f for all r > . More generally, if M is any Riemannian manifold and N = R , Willmore [Wil50] proved thatˆ S r f = f characterizes harmonic maps if and only if M is a harmonic manifold .The central theorem of this subsection is the following: Theorem 5.8.
Let f : M → N be a smooth map. For all x ∈ M , as r → : d (cid:18) S r f ( x ) , exp f ( x ) (cid:18) r m τ ( f ) x (cid:19) (cid:19) = O ( r ) (37) d (cid:18) B r f ( x ) , exp f ( x ) (cid:18) r ( m + ) τ ( f ) x (cid:19) (cid:19) = O ( r ) . (38)The following “generalized mean property for harmonic functions between Riemannian mani-folds” is an immediate corollary of Theorem 5.8: Theorem 5.9.
Let f : M → N be a smooth map. The following are equivalent:(i) f is harmonic.(ii) d ( f ( x ) , S r f ( x )) = O ( r ) for all x ∈ M .(iii) d ( f ( x ) , B r f ( x )) = O ( r ) for all x ∈ M .Remark . It is an immediate consequence of Proposition 5.6 that Theorem 5.8 and Theorem 5.9also hold for ˆ S r f ( x ) instead of S r f ( x ) , and ˆ B r f ( x ) instead of B r f ( x ) .In the remainder of this subsection we show Theorem 5.8. We only prove (37); the proof of(38) follows exactly the same lines. Consider a smooth map f : ( M , g ) → ( N , h ) and fix x ∈ M . Lemma 5.11.
Let r > . Denote by S r the Euclidean sphere of radius r > in T x M and σ r itsarea density. Then, as r → , the following estimate holds: ( S r ) ∫ S r exp − f ( x ) ◦ f ◦ exp x ( u ) d σ r ( u ) = r m τ ( f ) x + O ( r ) (39) where m = dim M . roof. We write the Taylor expansion of the function ˆ f = exp − f ( x ) ◦ f ◦ exp x : T x M → T f ( x ) N :ˆ f ( u ) = ˆ f ( ) + ( D ˆ f ) ( u ) + ( D ˆ f ) ( u , u ) + ( D ˆ f ) ( u , u , u ) + O ((cid:107) u (cid:107) ) . Let us integrate this identity over S r . We have:• ˆ f ( ) =
0, so ( S r ) ∫ S r ˆ f ( ) d σ r ( u ) = ( D ˆ f ) ( u ) = ( d f ) x ( u ) is an odd function of u , so ( S r ) ∫ S r ( D ˆ f ) ( u ) d σ r ( u ) = ( D ˆ f ) ( u , u ) = ( Hess f ) x ( u , u ) by definition of Hess f .Moreover, since this is a quadratic function of u , we can apply Lemma 5.12:1Area ( S r ) ∫ S r ( D ˆ f ) ( u , u ) d σ r ( u ) = r m tr (( Hess f ) x ) = r m τ ( f ) x . • ( D ˆ f ) ( u , u , u ) is an odd function of u , so ( S r ) ∫ S r ( D ˆ f ) ( u , u , u ) d σ r ( u ) = (cid:3) The following lemma is required to complete the proof of Lemma 5.11:
Lemma 5.12.
Let ( V , g = (cid:104)· , ·(cid:105)) be a Euclidean vector space and let B : V × V → R be a symmetricbilinear form. Denote by S r = S ( , r ) the sphere centered at the origin in V with radius r > , d σ r the area density on S r induced from the metric g and Area ( S r ) = ∫ S r d σ r its area. Then: ( S r ) ∫ S r B ( x , x ) d σ r = r dim V tr g ( B ) . Here we have denoted by tr g ( B ) the g -trace of B , i.e. the trace of the g -self adjoint endomorphismof V associated to B , or, equivalently, the trace of a matrix representing B in a g -orthonormal basis. Proof.
Let ( e , . . . , e n ) be basis of V which is g -orthonormal and B -orthogonal (the existence ofsuch a basis is precisely the spectral theorem). Let λ k = B ( e k , e k ) for k ∈ { , . . . n } . For any vector x = (cid:205) nk = x k e k , the quadratic form is given by B ( x , x ) = (cid:205) nk = λ k x k , hence: ∫ S r B ( x , x ) d σ r = n (cid:213) k = λ k ∫ S r x k d σ r . The integrals I k = ∫ S r x k d σ r can be swiftly computed starting with the observation that any twoof them are equal. Indeed, for k (cid:44) l , one can easily find a linear isometry ϕ such that x k ◦ ϕ = x l ;the change of variables theorem ensures that I k = I l . One can then write I k = n (cid:205) nl = I l for any k .That is I k = n ∫ S r (cid:0)(cid:205) nk = x k (cid:1) d σ r . However (cid:205) nk = x k = g ( x , x ) = r for any x ∈ S r . This yields I k = n ∫ S r r d σ r = r n Area ( S r ) . The desired result follows. (cid:3) It is easy to see that Theorem 5.8 follows immediately from Lemma 5.11 when N = R n . When N is not Euclidean, centers of mass in N are only defined implicitly (by equation (36)), so we haveto work harder to prove Theorem 5.8. The trick is to use Lemma 5.3.First we need a Riemannian geometry estimate in the following general setting. Let A , B , C be three points in a Riemannian manifold ( M , g ) . We assume that B and C are contained in asufficiently small ball centered at A for what follows to make sense. Denote by (cid:174) u A = ( exp A ) − ( B ) , (cid:174) u B = ( exp B ) − ( C ) , and (cid:174) u C = ( exp C ) − ( A ) . If we were in a Euclidean vector space, we could write: (cid:174) u A + (cid:174) u B + (cid:174) u C = .
35e would like to find an approximate version of this identity in general. Note that the sum (cid:174) u A + (cid:174) u B + (cid:174) u C does not make sense, because these vectors are based at different points. Denote (cid:174) v the parallel transport of − (cid:174) u C along the geodesic segment [ C , A ] and (cid:174) w the parallel transport of (cid:174) u B along the geodesic segment [ B , A ] . Let us also write (cid:174) u = (cid:174) u A for aesthetics. Now the vectors (cid:174) u , (cid:174) v , (cid:174) w are all based at A , and one expects that (cid:174) w = (cid:174) v − (cid:174) u up to some error term. Lemma 5.13.
Using the setting and notations above, we have as (cid:107) (cid:174) u (cid:107) → and (cid:107)(cid:174) v (cid:107) → : (cid:174) w = (cid:174) v − (cid:174) u + O ((cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) + (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v (cid:107) ) . We prove in fact the following more precise lemma:
Lemma 5.14.
Let ( M , g ) be a Riemannian manifold, fix A ∈ M . Let (cid:174) U and (cid:174) V be two tangentvectors at A , denote B ( t ) = exp A ( t (cid:174) U ) and C ( s ) = exp A ( s (cid:174) V ) . Let (cid:174) w ( t , s ) be the parallel transport of (cid:174) u B (cid:66) exp − B ( t ) ( C ( s )) along the geodesic segment from B ( t ) to A . Then: (cid:174) w ( t , s ) = s (cid:174) V − t (cid:174) U − t s R ( (cid:174) V , (cid:174) U ) (cid:174) U − ts R ( (cid:174) U , (cid:174) V ) (cid:174) V + O ( t + t s + t s ) . where R is the Riemann curvature tensor of ( M , g ) .Proof. First let us quickly discuss some general Riemannian geometry estimates in normal coordi-nates. We refer to [Bre96, Bre09] for more details on the computations that follow.In normal coordinates at a point A , the Riemannian metric g has the Taylor expansion g ij = δ ij − R ik jl x k x l + O (| x | ) where R ijkl is the Riemann curvature tensor at A , or rather its purely covariant version (this well-known fact of Riemannian geometry goes back to Riemann’s 1854 habilitation [Rie13]). One canderive from the expression for the Christoffel symbols Γ k ij = g kl ( g li , j + g l j , i − g ij , l ) that Γ k ij = − ( R k ijl − R k jil ) x l + O (| x | ) . One can then find the Taylor expansion of any geodesic x ( s ) , say with initial endpoint x = x ( ) andinitial velocity v , by solving the geodesic equation d x k d s + Γ kij d x i d s d x j d s =
0. One finds: x k ( s ) = x k + s v k − s R k il j v i v j x l + O ( s | x | ) . (40)We can rewrite (40) as a coordinate-free expression (but still in the chart given by exp A ) as x ( s ) = x + s v − s R ( x , v ) v + O ( s | x | ) . (41)One can also compute the parallel transport of a vector v along a radial geodesic x ( t ) = t x bysolving the parallel transport equation d v k d t + Γ k ij ( x ( t )) v i d x j d t . One finds that v k ( t ) = v k + R k jil v i x j ( t ) x l ( t ) + O ( t | x | ) , which we can rewrite as v ( t ) = v + R ( v , x ) x + O ( t | x | ) . (42)36et us now come back to the setting of Lemma 5.14. We shall work (implicitly) in the chartgiven by exp A . Note that we can write B = t (cid:174) U and C = s (cid:174) V in this chart. Let us denote by x (·) theunit speed geodesic from B to C , so that x ( ) = B , x ( r ) = C where r = d ( B , C ) , and x (cid:48) ( ) = (cid:174) U B isthe unit vector such that exp B ( r (cid:174) U B ) = C . By (41) we can write x ( r ) = x ( ) + r (cid:174) U B − r R ( x ( ) , (cid:174) U B ) (cid:174) U B + O ( r | x ( )| ) . In other words, recalling that x ( ) = B = t (cid:174) U and x ( r ) = C = s (cid:174) V , we have s (cid:174) V − t (cid:174) U = r (cid:174) U B − tr R ( (cid:174) U , (cid:174) U B ) (cid:174) U B + O ( rt ) . (43)On the other hand, the parallel transport of (cid:174) u B = r (cid:174) U B back to the origin along the radial geodesic [ A , B ] is given by, according to (42): (cid:174) w = r (cid:174) U B − t r R ( (cid:174) U B , (cid:174) U ) (cid:174) U + O ( t r ) . (44)Comparing (43) and (44), we see that (cid:174) w = s (cid:174) V − t (cid:174) U − t r R ( (cid:174) U B , (cid:174) U ) (cid:174) U − tr R ( (cid:174) U , (cid:174) U B ) (cid:174) U B + O ( t r ) . (45)Finally, let’s work to have s ’s and (cid:174) V ’s appear in this equation instead of r ’s and (cid:174) U B ’s. First note that (cid:174) rU B = s (cid:174) V − t (cid:174) U + O ( tr + rt ) according to (43), so using the fact that R ( (cid:174) U , (cid:174) U ) =
0, one can write: r R ( (cid:174) U B , (cid:174) U ) (cid:174) U = sR ( (cid:174) V , (cid:174) U ) (cid:174) U + O ( tr + rt ) r R ( (cid:174) U , (cid:174) U B ) (cid:174) U B = s R ( (cid:174) U , (cid:174) V ) (cid:174) V − tsR ( (cid:174) U , (cid:174) V ) (cid:174) U + O ( tsr + t sr + t r + t sr + t r ) . We thus get in lieu of (45): (cid:174) w = s (cid:174) V − t (cid:174) U − t s R ( (cid:174) V , (cid:174) U ) (cid:174) U − ts R ( (cid:174) U , (cid:174) V ) (cid:174) V + O ( t r + t sr ) . The conclusion follows, noting that r = O ( t + s ) by the triangle inequality. (cid:3) Remark . A direct consequence of Lemma 5.14 is the expansion of the distance squared: d ( exp A ( t (cid:174) U ) , exp A ( s (cid:174) V )) = (cid:107) s (cid:174) V − t (cid:174) U (cid:107) − R ( U , V , V , U ) s t + O (( t + s ) ) . The same formula has been observed by other authors, see e.g. [Raz15].We are now ready to wrap up the proof of Theorem 5.8:
Proof of Theorem 5.8.
Let u ∈ S r ⊂ T x M , and consider the triangle in N with vertices A (cid:66) f ( x ) , B (cid:66) T r f ( x ) , and C ( u ) (cid:66) f ( exp x ( u )) in N . With the notations introduced above Lemma 5.13,note that (cid:174) u = (cid:174) u A = r m τ ( f ) x , (cid:174) v ( u ) = exp − f ( x ) ( f ( exp x ( u ))) , and (cid:174) w ( u ) = P ( (cid:174) u B ( u )) where (cid:174) u B ( u ) = exp − T r f ( x ) ( f ( exp x ( u ))) and P : T A N → T B N is the parallel transport along the geodesic segment [ A , B ] . By Lemma 5.13, we have P ( (cid:174) u B ( u )) = (cid:174) w ( u ) = (cid:174) v ( u ) − (cid:174) u + O ((cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v ( u )(cid:107) + (cid:107) (cid:174) u (cid:107) (cid:107)(cid:174) v ( u )(cid:107) ) . (46)37ecause we have (cid:107) (cid:174) u (cid:107) = O ( r ) and (cid:107)(cid:174) v ( u )(cid:107) = O ( r ) , (46) may be rewritten: P ( (cid:174) u B ( u )) = (cid:174) v ( u ) − (cid:174) u + O ( r ) . (47)We now integrate (47) over u ∈ S r :1Area ( S r ) ∫ S r P ( (cid:174) u B ( u )) d σ r ( u ) = ( S r ) ∫ S r (cid:16) (cid:174) v ( u ) d σ r ( u ) − (cid:174) u + O ( r ) (cid:17) , which we can rewrite as P (cid:18) ( S r ) ∫ S r (cid:174) u B ( u ) d σ r ( u ) (cid:19) = (cid:18) ( S r ) ∫ S r (cid:174) v ( u ) d σ r ( u ) (cid:19) − (cid:174) u + O ( r ) . Now, Lemma 5.11 says precisely that (cid:16) ( S r ) ∫ S r (cid:174) v ( u ) d σ r ( u ) (cid:17) = (cid:174) u + O ( r ) . We thus get ( S r ) ∫ S r (cid:174) u B ( u ) d σ r ( u ) = P − ( O ( r )) = O ( r ) . That is, ∫ S r exp − T r f ( x ) ( f ( exp x ( u ))) d σ r ( u ) = O ( r ) . Recalling that S r f ( x ) is by definition the center of mass of the function u ∈ S r (cid:55)→ f ( exp x ( u )) , wecan apply Lemma 5.3 to conclude that d ( S r f ( x ) , T r f ( x )) = O ( r ) . (cid:3) We now discuss center of mass methods as an alternative to the heat flow in order to minimize theenergy functional. The basic idea is to iterate the process of replacing a function f : M → N byits average on balls (or spheres) of radius r >
0, hopefully converging to a map f ∗ r that is almostharmonic when r is small. Observe that Theorem 5.8 shows that this method is very close to aconstant step gradient flow for the energy functional, i.e. an Euler method with fixed stepsize.The next proposition is claimed in [Jos97, Lemma 4.1.1]. Proposition 5.16.
Let ( M , µ ) be a measure space, let ( N , d ) be a Hadamard metric space and let η : M × M → [ , + ∞) be a measurable symmetric function. Define the Jost energy functional by E ( f ) = ∫ M ∫ M η ( x , y ) d ( f ( x ) , f ( y )) d µ ( y ) d µ ( x ) . (48) For a measurable map f : M → N , let ϕ ( f ) : M → N be the map such that for all x ∈ M , ϕ ( f )( x ) is the center of mass of f for the measure η ( x , ·) µ . Then for every f with finite energy we have: E ( ϕ ( f )) (cid:54) E ( f ) . (49) Moreover, the following are equivalent:(i) Equality holds in (49) .(ii) ϕ ( f ) = f almost everywhere in ( M , µ ) .(iii) f is a minimizer of E . enter of mass method in the smooth setting Now assume that M and N are both Riemannian manifolds. For r > η r ( x , y ) described in § 1.2, so that E r is the r -approximate energy. The map ϕ ( f ) of Proposition 5.16 isthen the same as the map ˆ B r f introduced in Definition 5.4. It is tempting to iterate the processof averaging f in order to try and minimize E r . The next theorem guarantees the success of thismethod under suitable conditions. Moreover, recall that the energy functional E on L ( M , N ) is the Γ -limit of E r as r → E r converge to minimizersof E (possibly up to subsequence). Theorem 5.17.
Let M and N be Riemannian manifolds, assume N is compact and with nonpositivesectional curvature. In any homotopy class of continuous maps M → N where the r -approximateenergy E r admits a unique minimizer f ∗ , the sequence ( f k ) k ∈ N defined by f k + = ˆ B r f k convergeslocally uniformly to f ∗ for any choice of a locally Lipschitz continuous f .Proof. We reduce the proof to a combination of Lemma 5.18 and Lemma 5.19 below. Denote by X the connected component of f in C( M , N ) , let E : X → R denote the restriction of E r , and let ϕ : X → X be the map f (cid:55)→ ˆ B r f (it is easy to see that ϕ preserves X ). Lemma 5.18 guaranteesimmediately that the sequence ( f k ) k ∈ N is equicontinuous. Since N is compact, it follows from theArzelà-Ascoli theorem that the sequence ( f k ) k ∈ N is relatively compact in X for the compact-opentopology. By Proposition 5.16 and the assumption that f ∗ is unique, we have E ( ϕ ( f )) (cid:54) E ( f ) forall f ∈ X , with equality only if f = f ∗ . Conclude by application of Lemma 5.19. (cid:3) Lemma 5.18.
Let f : M → N where M and N are Riemannian manifolds, with N complete andnonpositively curved. If f is locally Lipschitz continuous, then so is ˆ B r f . Moreover, the Lipschitzconstant of ˆ B r f is bounded above by the Lipschitz constant of f on any compact K ⊆ M .Proof. For simplicity, we assume that f is globally L -Lipschitz, and argue that ˆ B r f is also L -Lipschitz; the proof can easily be extended to the general case by restricting to compact sets. Firstwe assume that M is Euclidean, in fact let us put M = R m . Let x , y ∈ M , write y = x + h so that d ( x , y ) = (cid:107) h (cid:107) . By definition, ˆ B r f ( y ) is the point of N such that1vol ( B ( y , r )) ∫ B ( y , r ) exp − B r f ( y ) ( f ( v )) d v g ( v ) = . (50)Note that the map u (cid:55)→ u + h defines an isometry from B ( x , r ) to B ( y , r ) . Making the change ofvariables v = u + h , we derive from (50): ∫ B ( x , r ) exp − B r f ( y ) ( f ( u + h )) d v g ( u ) = . It follows that ∫ B ( x , r ) exp − B r f ( y ) ( f ( u )) d v g ( u ) = ∫ B ( x , r ) (cid:104) exp − B r f ( y ) ( f ( u )) − exp − B r f ( y ) f ( u + h )) (cid:105) d v g ( u ) . (51)Assume without loss of generality that N is simply connected (one can lift to the universal cover).Then N is a Hadamard manifold and in particular a CAT ( ) metric space, which implies that forany p ∈ N , the map exp − p : N → T p N is distance nonincreasing (in fact, the converse is also true).We can therefore derive from (51): (cid:13)(cid:13)(cid:13)(cid:13)∫ B ( x , r ) exp − B r f ( y ) ( f ( u )) d v g ( u ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:54) ∫ B ( x , r ) d ( f ( u ) , f ( u + h )) d v g ( u ) . (52)39t follows from (52) and the fact that f is L -Lipschitz that (cid:13)(cid:13)(cid:13)(cid:13) ( B ( x , r )) ∫ B ( x , r ) exp − B r f ( y ) ( f ( u )) d v g ( u ) (cid:13)(cid:13)(cid:13)(cid:13) (cid:54) L (cid:107) h (cid:107) . (53)Lemma 5.3 now applies directly to (53) to conclude that d ( ˆ B r f ( x ) , ˆ B r f ( y )) (cid:54) L (cid:107) h (cid:107) . Since (cid:107) h (cid:107) = d ( x , y ) , we have shown that ˆ B r f is L -Lipschitz, as desired.Now we argue that the argument extends to the case where M is an arbitrary Riemannianmanifold using a local to global trick. First note that a function is globally L -Lipschitz if and onlyif it is locally L -Lipschitz. Here we mean by locally L -Lipschitz the property that for any x ∈ M ,there exists δ > d ( y , x ) < δ implies d ( f ( y ) , f ( x )) (cid:54) Ld ( y , x ) . We leave it to the readerto show that in any path metric space, locally L -Lipschitz in this sense implies globally L -Lipschitz.With this observation in mind, let us finish the proof. The key argument that worked abovewhen M is Euclidean is that there exists an isometry from B ( x , r ) to B ( y , r ) that displaces everypoint of at most d ( x , y ) . This is no longer true when M is an arbitrary Riemannian manifold,however note that it is almost true when x and y are very close. Quantifying this properly, clearlyone can show that for every x ∈ M and for every L (cid:48) > L , there exists δ > d ( y , x ) < δ implies d ( ˆ B r f ( y ) , ˆ B r f ( x )) (cid:54) L (cid:48) d ( y , x ) . Thus we have shown that ˆ B r f is locally L (cid:48) -Lipschitz, andtherefore globally L (cid:48) -Lipschitz. Since this is true for all L (cid:48) > L , ˆ B r f is actually L -Lipschitz. (cid:3) Lemma 5.19.
Let X be a first-countable topological space and let E : X → R a continuous functionthat admits a unique minimizer x ∗ . Assume that ϕ : X → X is a continuous map such that:(i) E ( ϕ ( x )) (cid:54) E ( x ) for all x ∈ X , with equality only if x = x ∗ .(ii) For all x ∈ X , the set { ϕ k ( x ) , k ∈ N } is relatively compact in X .Then for any x ∈ X , the sequence ( ϕ k ( x )) k ∈ N converges to x ∗ .Proof. In any topological space, in order to show that a sequence ( x k ) k ∈ N converges to a point x ∗ ,it is enough to show that:(a) The sequence ( x k ) has no cluster points except possibly x ∗ .(b) Any subsequence of ( x k ) admits a cluster point.Indeed, assume that ( x k ) does not converge to x ∗ , then there exists a subsequence of ( x k ) that avoidsa neighborhood of x ∗ . This subsequence must have a cluster point by (b), which cannot be x ∗ .However this point is also a cluster point of the sequence ( x k ) , contradicting (a).Coming back to Lemma 5.19, let x ∈ X and denote x k = ϕ k ( x ) . The sequence ( x k ) satisfies(b) because of the assumption (ii). So we need to show that ( x k ) satisfies (a) and we are done. Let y be a cluster point of ( x k ) , we need to show that y = x ∗ . Since X is first-countable, there exists asubsequence ( x k n ) n ∈ N converging to y . Observe that by assumption (i), since k n (cid:54) k n + (cid:54) k n + , E ( x k n + ) (cid:54) E ( x k n + ) (cid:54) E ( x k n ) . (54)By continuity of E , we have lim E ( x k n ) = lim E ( x k n + ) = E ( y ) , so (54) implies that lim E ( x k n + ) = E ( y ) . On the other hand, since x k n + = ϕ ( x k n ) and ϕ is continuous, we have lim x k n + = ϕ ( y ) , solim E ( x k n + ) = E ( ϕ ( y )) . Thus E ( ϕ ( y )) = E ( y ) , and we conclude that y = x ∗ by (i). (cid:3) Discrete center of mass method
We now prove that Theorem 5.17 also holds in the discrete setting developed in § 2, providingan alternative method to the discrete heat flow discussed in § 4.2. Let G be a biweighted ˜ S -triangulated graph (see § 2.2), let N be a Hadamard manifold and let ρ : π S → Isom ( N ) be a grouphomomorphism. We recall that the discrete energy functional E G : Map eq (G , N ) → R coincideswith Jost’s energy functional (48) for the appropriate choice of kernel η (Proposition 2.19).40otivated by Proposition 5.16, we note that in this discrete setting the measure η ( x , ·) µ is givenby the weighted atomic measure (cid:213) y ∼ x ω xy µ ( x ) δ y , (55)where δ y is the Dirac measure at y . Now the averaging map f (cid:55)→ ˆ B r f takes the following form: Definition 5.20.
The discrete center of mass method on Map eq (G , N ) is given by f (cid:55)→ ϕ ( f ) , where ϕ ( f )( x ) is the center of mass of f for the atomic measure (55).Note that Proposition 5.16 applies in this setting (cf. Proposition 2.22). Under certain assump-tions on N and ρ , we obtained strong convexity of E G in Theorem 3.20, so that, in particular, E G has a unique minimum. The same assumptions have similarly useful consequences here: Theorem 5.21.
Let N be a manifold of negative curvature bounded away from and let ρ be afaithful representation whose image is contained in a discrete subgroup of Isom ( N ) acting freely,properly, and cocompactly on N . Given any initial discrete equivariant map f ∈ Map eq (G , N ) , thediscrete center of mass method converges to the unique discrete harmonic map.Proof. The proof is a similar but easier version of the proof of Theorem 5.17. Let f ∈ Map eq (G , N ) ,and define the sequence ( f k ) k ∈ N by f k + = ϕ ( f k ) . The assumption on ρ means that we can work ina compact quotient of N , making the sequence ( f k ) pointwise relatively compact. Since the actionof π S on G is cofinite, it is easy to see that the condition that the family { f k } is equicontinuousis vacuous. Hence the family { f k } is relatively compact in Map eq (G , N ) . Since Proposition 5.16holds in this setting, all the requirements are met to conclude with Lemma 5.19. (cid:3) Remark . In [JT07], Jost-Todjihounde describe an iterative process to obtain a discrete harmonicmap from an edge-weighted triangulated graph G to a target space that admits centers of mass (e.g.Hadamard spaces). Theorem 5.21 can be viewed as a strengthened version of their result in tworespects: For one, we avoid Jost-Todjihounde’s passage to a subsequence of ( f k ) . Moreover, Jost-Todjihounde start by subdividing G and pursuing centers of mass in two phases, separately forvertices and midpoints of edges. Our discrete center of mass method requires no such subdivision. cosh -center of mass Theorem 5.21 provides an effective method to compute discrete equivariant harmonic maps, al-ternative to the discrete heat flow (see § 4.2), as long as one is able to compute centers of mass.Unfortunately, non-Euclidean centers of mass are not easily accessible. Even finding the barycenterof three points in the hyperbolic plane is a nontrivial task. While it is possible to use gradientdescent method (see [ATV13]), it is computationally expensive and, in any case, not possible to doprecisely in finite time. We present a clever variant to barycenters, well-suited to hyperbolic space H n , that avoids this issue. We thank Nicolas Tholozan for bringing this idea to our attention. Definition 5.23.
Let ( Ω , F , µ ) be a probability space, ( X , d ) be a metric space, and h : Ω → X ameasurable map. A cosh -center of mass of h is a minimizer of the function P h : X → R x (cid:55)→ ∫ Ω ( cosh d ( x , h ( y )) − ) d µ ( y ) . X is a Riemannian manifold, a cosh-center of mass G is characterized by ∫ Ω sinhc d ( G , h ( y )) exp − G ( h ( y )) d µ ( y ) = , (56)where sinhc ( x ) = sinh ( x )/ x is the cardinal hyperbolic sine function.Equation (56) implies that if supp ( h ∗ µ ) is contained in a strongly convex region U (e.g. a ball ofsmall enough radius), then any cosh-center of mass is contained in U as well: if x is outside U , theneach vector exp − x ( h ( y )) , for y ∈ supp ( h ∗ µ ) , is contained in an open half-space in T x X containingexp − x ( U ) , and (56) cannot be satisfied.Let us now specialize to the case where X = H n is the hyperbolic n -space. In this setting thefunction F ( x ) = cosh ( d ( x , x )) − F ( x ) = sinhc ( d ( x , x )) exp − x ( x ) Hess ( F ) x ( v , v ) = F ( x )(cid:107) v (cid:107) . In particular, F is a strongly convex function on H n with modulus of strong convexity α = h ∈ L ( Ω , H n ) quickly follows.The main advantage of the cosh-center of mass is that it admits an explicit description, muchlike the Euclidean barycenter. For this we work in the hyperboloid model for H n , i.e. H = { x ∈ R n , : (cid:104) x , x (cid:105) = − , x n + > } , where Minkowski space R n , is defined as R n + equipped with the indefinite inner product (cid:104) x , y (cid:105) = x y + · · · + x n y n − x n + y n + . This inner product induces a Riemannian metric on H of constant curvature − Proposition 5.24.
The cosh -center of mass in H n ≈ H is equal the orthogonal projection of theEuclidean barycenter in Minkowski space R n , to the hyperboloid H ⊂ R n , .Proof. We prove Proposition 5.24 for a finite collection of points for comfort; the generalization toany probability measure is immediate. Consider points p , . . . , p n ∈ H with weights w , . . . , w n satisfying (cid:205) i w i =
1, let p be their Euclidean barycenter in R n , , and let q indicate the orthogonal(i.e. radial) projection of p to H . By (56), it suffices to check that (cid:213) i w i sinhc d ( q , p i ) exp − q ( p i ) = . (57)Let P be the tangent plane to H at q , i.e. the affine plane in R n , which is orthogonal to the line R q through q . The orthogonal projection π : R n , → P is an affine map, so the identity (cid:205) i w i ( p i − p ) = (cid:205) i w i ( q i − q ) = P , where q i = π ( p i ) . It is straightforward to compute q = p √ −(cid:104) p , p (cid:105) and q i = p i + q + (cid:104) p i , p (cid:105)−(cid:104) p , p (cid:105) p .Geodesics in the hyperboloid are intersections of 2-dimensional subspaces of R n , with H , so q i − q is a vector in T q H pointing towards p i , and we can compute its length: (cid:107) q i − q (cid:107) = (cid:104) p i , q (cid:105) − = sinh d H ( q , p i ) Thus we proved that q i − q = sinh d H ( q , p i ) d H ( q , p i ) exp q − ( q i ) and we get (57) as desired. (cid:3) Proposition 5.25.
If a probability measure is supported in a ball of radius r , then its center of mass p and its cosh -center of mass q and are within O ( r ) of each other.Proof. Assume µ has finite support { p , . . . , p n } for comfort and denote w i = µ ({ p i }) the weights.By (56) we can write: (cid:213) i w i exp − q ( p i ) = (cid:213) i w i ( − sinhc d ( p i , q )) exp − q ( p i ) . (58)Because q must be contained in the same ball of radius r as { p i } , we find that d ( p i , q ) < r for each i . Given that sinhc is a nondecreasing function we derive from (58) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:213) i w i exp − q ( p i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:54) (cid:213) i w i ( sinhc d ( p i , q ) − ) (cid:13)(cid:13) exp − q ( p i ) (cid:13)(cid:13) (cid:54) (cid:213) i w i ( sinhc ( r ) − ) · ( r ) = r ( sinhc ( r ) − ) . Lemma 5.3 now implies that d ( p , q ) (cid:54) r ( sinhc ( r ) − ) . The conclusion follows, since 2 r ( sinhc ( r ) − ) = r + O ( r ) . (cid:3) Note that we did not use in the proof of Proposition 5.25 that we are working in H n : thisproposition holds in any Riemannian manifold. Harmony
Harmony is a computer program developed by the first two authors (Jonah Gaster and Brice Loustau).It is a cross-platform software with a graphical user interface written in C ++ code using the Qtframework. In its current state, it totals about 14 ,
000 lines of code.
Harmony is a free and open source software under the GNU General Public License. It isavailable on GitHub at https://github.com/seub/Harmony .For more information, including install instructions and a quick guide to get started using thesoftware, please visit the dedicated web site: . We provide a brief overview of
Harmony ’s algorithms allowing effective computation of discreteequivariant harmonic maps H → H with respect to a pair of Fuchsian representations. A flowchartshowing how the main algorithms fit into the program is pictured in Figure 12.We fix an identification of the closed oriented topological surface S of genus g as P /∼ , where P is a topological 4 g -gon with oriented sides labelled a , b , a − , b − , a , etc..43e parametrize hyperbolic structures on S using the famous Fenchel-Nielsen coordinates. Thisrequires choosing a pants decomposition of S . Harmony is equipped to make such choices forarbitrary g in a way that minimizes future error propagation.The input is a pair of Fenchel-Nielsen coordinates for hyperbolic structures X and Y on S ,the domain and target hyperbolic surfaces respectively. These can be entered by the user in a‘Fenchel-Nielsen selector’ window: see Figure 7. Step 1: Construct the fundamental group and pants decomposition
After getting the genus g as input, Harmony constructs the fundamental group of the surface as anabstract structure. It then chooses a pants decomposition of the surface, yielding a decompositionof the fundamental group in terms of amalgamated products and HNN extensions of fundamentalgroups of pairs of pants. This is done recursively on the genus using a binary tree structure.
Step 2: Construct representations ρ X and ρ Y This step performs the translation of Fenchel-Nielsen coordinates to Fuchsian representations.
Harmony starts by computing the representation of the fundamental group of each pair of pantsusing formulas that can be found in e.g. [Kou94, Prop. 2.3] or [Mas99, Mas01]. It then computesthe representation of the whole fundamental group using its decomposition discussed in Step 1.
Step 3: Construct fundamental domains P X and P Y This step computes polygonal fundamental domains P X and P Y in H for the Fuchsian groups inthe images of ρ X and ρ Y . These polygons should be ‘as convex as possible’ in order to ensure goodbehavior of the discrete heat flow.Both P X and P Y come with π S -equivariant identifications to the topological 4 n -gon P thatrecord side pairings. Because the vertices of P are all in the same π S -orbit, P X is determined by asingle point in H . With this combinatorial setup, a best choice of polygon is obtained minimizingan adequate cost function F : H → R + . This is done with a straightforward Newton method. Step 4: Construct a triangulation of P X This step computes a triangulation of the fundamental domain P X . Finer and finer meshes can thenbe obtained by subdivision (see Definition 2.3). As explained in [GLM19], it is crucial to keep thesmallest angle of the triangulation as large as possible.Unfortunately, P X already typically has very small angles. In order to avoid subdividing theseangles further, we first introduce new Steiner vertices evenly spaced along the sides of P X . Theresulting polygon is triangulated with a greedy recursive algorithm maximizing the smallest angle.See Figure 7 for a sample output. Remark . The algorithm seems to always produce acute triangulations, a necessary condition forthe definition of the edge weights (see § 2.1), but we do not know whether this always holds. Acutetriangulations of surfaces are part of a fascinating area of current research [CdVM90, Zam13].44 tep 5: Construct the ρ X -invariant mesh M The mesh M consists of a list of meshpoints M ( ) ⊂ H , each of which is equipped with a listof references to its neighboring meshpoints, and possibly side-pairing information. This data isinitially recorded from the triangulated polygon P X . Then, given a user chosen mesh depth k (cid:62) M is replaced with the k th iterated midpoint refinement of M (see Definition 2.3). Remark . Constructing the adequate data structure to efficiently store the mesh data is a difficultchallenge: the corresponding C ++ classes are the most sophisticated in the code of Harmony . Step 6: Initialize an equivariant map
The triangulation of P X may be transported to one for P Y via the π S -equivariant maps P X ≈ P ≈ P Y . Because the mesh is built from midpoint refinements, this identification provides an initial discreteequivariant map f : H → H . A sample initial map is showed in Figure 8. Step 7: Run the discrete flow
Harmony is ready to run: either the discrete heat flow with fixed or optimal stepsize (see § 4.2) orthe cosh-center of mass method (§ 5.3, § 5.4).
Harmony uses several threads so that the flow isdisplayed “live” as it is being computed. See Figure 9 for a screenshot of
Harmony mid-flow.The flow is iterated until the error reaches a preset tolerance, or when it is stopped by the user.Both the discrete heat flow and the center of mass method typically converge very well. Refer to§ 4.3 for a comparison of the methods. Figure 10 shows a sample output equivariant harmonic map.
In this final subsection we briefly explain how
Harmony provides visual confirmation of a qualitativephenomenon that is well-known in Teichmüller theory. We hope that future development of theprogram will allow many more experimental investigation of theoretical aspects. We refer to[DW07] for more details on what follows.As mentioned in § 1.5, taking the Hopf differential of harmonic maps allows one to parametrizeTeichmüller space by holomorphic quadratic differentials. More precisely, given a closed oriented S of genus (cid:62)
2, let
F ( S ) denote the Fricke-Klein space of S , i.e. the deformation space of hyperbolicstructures on S up to isotopically trivial diffeomorphisms. Fix a complex structure X on S , anddenote Q ( X ) the vector space of holomorphic quadratic differentials on X . For any σ ∈ F ( S ) , thereis a unique harmonic map f σ : X → ( S , σ ) . Taking its Hopf differential yields a map H : F ( S ) → Q ( X ) σ (cid:55)→ ϕ f σ . Wolf [Wol89] proved that the map H is a global diffeomorphism from F ( S ) to Q ( X ) . Furthermore, H continuously extends to the “boundaries at infinity”: ∂ H : ∂ F ( S ) → ∂ Q ( X ) . (59)45ere the boundary ∂ F ( S ) compactifying Fricke-Klein space is the Thurston boundary ∂ F ( S ) (cid:66) PMF ( S ) , the projective space of measured foliations on S . The boundary of the vector space Q ( X ) is simply its projectivization: ∂ Q ( X ) (cid:66) P ( Q ( X )) . It can be identified to the projectivespace of measured laminations PML( S ) by assigning to any holomorphic quadratic differential itshorizontal foliation. Thus the boundary map ∂ H of (59) may be described as a map ∂ H : PMF ( S ) → PML( S ) . This map has a nice geometric interpretation as the well-known “pull tight” map which assigns toeach non-singular leaf of a measured foliation the unique geodesic in its homotopy class.Concretely, this means that a high energy harmonic map f : X → ( S , σ ) typically has a specificbehavior dictated by its Hopf differential ϕ , namely: the zeros of ϕ are blown up to ideal hyperbolictriangles, while the rest of the surface is compressed onto the measured lamination dual to ϕ . Thisbehavior is well verified by Harmony . As an example, Figure 11 shows the image of a high energyharmonic map: observe how the most contracted (darker) regions approach a geodesic lamination,while the most dilated (lighter) regions approach a union of ideal triangles.46 .4 Illustrations of
Harmony
Figure 6:
Harmony ’s main user interface.Figure 7: Fenchel-Nielsen coordinates selector.47igure 8: An initial discrete equivariant map f . The highlighted blue triangles are matched.Figure 9: Harmony mid-flow.48igure 10: Sample output harmonic map. The brighter central regions are fundamental domains.Figure 11: Sample output high energy equivariant harmonic map.49
NPUT Fenchel-Nielsencoordinates for X INPUT Fenchel-Nielsencoordinates for Y INPUT genusINPUT mesh depth INPUT flow methodINPUT displayparameters Construct π S = h a , b , . . . , a g , b g | Q [ a i , b i ] i Construct Fuchsian ρ X Construct fund domain P X Triangulate P X Construct Fuchsian ρ Y Construct fund domain P Y Construct ρ X -invariant mesh M : · graph data of M· triangulation data of M· metric data of M Are both ρ X and ρ Y set?Yes NoConstruct initial ( ρ X , ρ Y )-equivariantmap M → H using P X ≈ P Y Iterate flow: update map
M → H Is error < (cid:15) ? or user stop?Yes NoDisplay discrete harmonic map Figure 12: Flowchart representing
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