Deforming solutions of geometric variational problems with varying symmetry groups
aa r X i v : . [ m a t h . DG ] M a r DEFORMING SOLUTIONS OF GEOMETRIC VARIATIONALPROBLEMS WITH VARYING SYMMETRY GROUPS
RENATO G. BETTIOL, PAOLO PICCIONE, AND GAETANO SICILIANO
Abstract.
We prove an equivariant implicit function theorem for variationalproblems that are invariant under a varying symmetry group (correspondingto a bundle of Lie groups). Motivated by applications to families of geometricvariational problems lacking regularity, several non-smooth extensions of theresult are discussed. Among such applications is the submanifold problemof deforming the ambient metric preserving a given variational property of aprescribed family of submanifolds, e.g., constant mean curvature, up to theaction of the corresponding ambient isometry groups. Introduction
Geometric variational problems are typically invariant under a Lie group of sym-metries. As a potpourri of examples, closed geodesics γ : S → M in a Riemannian(or semi-Riemannian) manifold are invariant under rotation of the parameter; con-stant mean curvature (CMC) hypersurfaces x : N → M are invariant under isome-tries (i.e., rigid motions) of the ambient space M ; and harmonic maps φ : N → M between Riemannian manifolds are invariant under compositions with isometriesfrom both domain and target spaces. These symmetries create an inherent ambi-guity among solutions that, a priori, are different from the analytical viewpoint,but rather indistinguishable from the geometric viewpoint. Thus, in order to studydeformations of such geometric objects using analytical tools, it is essential to takeinto account the effect of symmetries. Deformation issues for families of geometricvariational problems whose group of symmetries remains unchanged were recentlystudied in [2], extending classic results of Dancer [7, 8]. The goal of the presentpaper is to address this issue for a much richer class of deformations, for which thegroup of symmetries varies , in an appropriate sense. Applications are discussed forall the above mentioned examples.The starting point in studying such deformations is to properly formalize theconcept of a smooth (or even continuous) family of symmetry groups G parametrizedby λ ∈ Λ. This is achieved considering bundles of Lie groups over Λ, which areLie groupoids whose source and target maps coincide, see for instance [10, 14, 16,17, 19]. An essential (and somewhat surprising) feature of these objects is that thetype of regularity that holds for bundles of Lie groups does not yield smoothnessof the global geometric structure of the group, or even continuity of the topologicalstructure, but rather smoothness of the group operations.
Date : July 17, 2018.2010
Mathematics Subject Classification.
As a motivation and companion example, let G λ be the identity component ofthe simply-connected n -dimensional space form of constant curvature λ . It is well-known that G λ is a Lie group of dimension n ( n + 1) and is isomorphic to SO ( n, R n ⋊ SO ( n ) or SO ( n +1), according to the cases λ < λ = 0 and λ > λ ∈ R , the corresponding groups G λ vary frombeing noncompact to compact as λ becomes positive. Nevertheless, G = { G λ : λ ∈ R } can still be proved to be a bundle of Lie groups (according to Definition 2.1).This is essentially due to the fact that the corresponding Lie algebras g λ form asmooth subbundle of gl ( n + 1), see Proposition 2.4 and Example 2.6. Such G is anexample of a non-locally trivial bundle of Lie groups over R . To our knowledge,this example was first studied in [24, 25], where isometry groups of other symmetricspaces were also considered.The main result of [25] provides deformations of a family of CMC immersionsof the 2-torus T into R to CMC immersions of T into S or H . The startingfamily of immersions into R , whose existence has been first proved by H. Wente[27], is of significant importance in Differential Geometry, as it provided a long-searched counter-example to a famous conjecture of H. Hopf. Smoothness of thefamily of ambient isometry groups when passing from positive to negative curvaturehas been employed in [25] to setup the framework for an implicit function theoremfor CMC immersions of T into 3-dimensional space forms, up to rigid motions.Similar situations in which the ambient space is deformed to yield a correspondingfamily of CMC (in this case, minimal ) submanifolds are found in [12, 20]. Thesecontributions provide, respectively, a deformation of minimal helicoids with handlesfrom S ( r ) × R to R as r → + ∞ , and of compact pieces of the Costa-Hoffman-Meeks surface in R to H × R . The purpose of this paper is to develop this type ofdeformation technique in a general variational context, dealing with perturbationsof geometric variational problems that entail a change of the symmetry group.Equipped with the above notion of varying symmetry groups, we now explainour main result. Consider a family of smooth functionals parametrized by λ ∈ Λ, f λ : X → R , f λ ( g λ · x ) = f λ ( x ) for all x ∈ X, g λ ∈ G λ , where X is a Banach manifold on which the bundle of Lie groups G = { G λ : λ ∈ Λ } acts by diffeomorphisms, i.e., X has a G λ -action for all λ ∈ Λ. We are interestedin solutions of the equation(1.1) d f λ ( x ) = 0 . Suppose the G -action on X is regular , in the sense that the map β λx : G λ → X, β λx ( g ) = g · x, is differentiable at the identity 1 λ ∈ G λ , for all λ ∈ Λ, and the section ( λ, x ) d β λx (1 λ ) is continuous. Denote by D λx ⊂ T x X the image of d β λx (1 λ ), which coincideswith the tangent space at x to its G λ -orbit. Note that since f λ is G λ -invariant,its linearization d f λ : X → T X ∗ is G λ -equivariant. In particular, if x ∈ X is acritical point of f λ , then the entire G λ -orbit of x consists of critical points of f λ . d β λx (1 λ ) is a section of the vector bundle Hom( g , T X ) over Λ × X , whose fiber over ( λ, x ) isthe space of linear maps from g λ to T x X . Denoting by γ λg : X → X the diffeomorphism X ∋ x g · x ∈ X , where g ∈ G λ , there is aaction of G λ on T X , given by g · v = d γ λg ( x ) v ∈ T gx X for all v ∈ T x X and g ∈ G λ ; and an inducedaction on the cotangent bundle T X ∗ , given by g · α = α ( g − · ) ∈ T gx X ∗ , for all α ∈ T x X ∗ . EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 3
Moreover, the kernel of the second derivative d f λ ( x ) contains D λx . The criticalpoint x is called equivariantly nondegenerate if the reverse inclusion also holds, i.e.,the kernel of d f λ ( x ) is the smallest possible. In this situation, our main abstractresult is the following equivariant implicit function theorem: Theorem.
Let ( λ , x ) ∈ Λ × X be a solution of (1.1) , where x is an equiv-ariantly nondegenerate critical point of f λ . Assume that the second variation d f λ ( x ) is represented by a self-adjoint Fredholm operator. Then, there existan open neighborhood U ⊂ Λ of λ , an open neighborhood V ⊂ X of x and a map U ∋ λ x λ ∈ V such that ( λ, x ) ∈ U × V is a solution of (1.1) if and only if x isin the G λ -orbit of x λ . A proof of the above result will be given in steps, according to the level of reg-ularity assumptions made on the group action. We stress that in many variationalproblems, especially those involving actions of diffeomorphism groups (e.g., prob-lems invariant under reparametrizations), the canonical symmetry group does notact smoothly . This subtle issue originates from the loss of differentiability causedby the chain rule, in an action by composition. A paradigmatic example of thissituation is given by the CMC embedding problem, where the ambient isometrygroup acts on unparameterized embeddings, see Subsection 5.1 and [2].A first proof of the above result, for the regular case, is given in Section 3.Here, the main issue concerns the appropriate notion of Fredholmness that must beused. Restricting to the realm of
Hilbert manifolds, where a standard Fredholmnessassumption for d f λ ( x ) would be feasible, is not suited for most geometric varia-tional problems that are naturally defined on Banach manifolds. In this situation,there is no clear Fredholmness condition that can be required for d f λ ( x ), giventhat in general there are no Fredholm operators from a Banach space to its dual. Inorder to deal with this problem, one often uses an auxiliary pre-Hilbert structureunderlying the functional space (typically, an L -pairing), relative to which onerequires Fredholmness (see Definition 3.1). From a functional analytical viewpoint,an interesting observation is that Fredholmness and symmetry with respect to anincomplete inner product in general does not imply the vanishing of the Fredholmindex, as it happens for self-adjoint operators on Hilbert spaces. This is discussedin Appendix A, where a criterion for the vanishing of Fredholm index of symmetricoperators is given in terms of a certain ellipticity condition.The essential ingredient for the proof in the regular case is the existence of asubmanifold of X which is transversal to the group orbit G λ ( x ) at x . As to thenonregular case, discussed in Section 4, the transversality argument is replaced by atopological degree argument, whose abstract formulation is given in Proposition 3.3.For such general result, it is only required that the group action is differentiablein a dense subset of X , which happens to be the case in a large class of geometricvariational problems. Roughly speaking, when the functional space consists of C k -submanifolds Σ of a smooth manifold M , and the action is given by applyingdiffeomorphisms of M , differentiability occurs at those Σ that are C k +1 .Several applications of the above abstract result are discussed in Section 5, in thecontext of geometric variational problems. First, we deal with the above mentionedperturbation problem of CMC hypersurfaces in families of Riemannian manifoldswith smoothly varying isometry groups. We also consider the case of immersed For details on these technical regularity assumptions, see Section 3.
R. G. BETTIOL, P. PICCIONE, AND G. SICILIANO submanifolds, possibly with boundary. Second, we discuss perturbations of har-monic maps between Riemannian manifolds, where the metrics vary on both targetand source manifolds. Finally, we consider a perturbation problem for Hamilton-ian stationary Lagrangian submanifolds of a symplectic manifold (
M, ω ). Here,we consider a family of Riemannian metrics on M whose isometry groups act in aHamiltonian fashion on ( M, ω ), obtaining a perturbation result of Lagrangian sub-manifolds that are Hamiltonian stationary with respect to a K¨ahler metric. Thisimproves a result recently obtained in [3].Finally, a brief comment on the smoothness of families of Lie groups is in order.We point out that, despite the fact that the general Lie algebroid and Lie groupoidtheories provide the existence of smooth families of Lie groups associated to bundlesof Lie algebras, this abstract result is not suited for the purposes of the presentpaper. Namely, we observe that this only provides bundles of simply-connected Liegroups, unrelated to each other, except for the smoothness condition. Moreover, theresult cannot be employed in order to establish whether a given family of (possiblynot simply-connected) Lie groups is smooth, which is the central point that mustbe addressed here. This suggests the formulation of a smoothness criterion forfamilies of Lie subgroups of a fixed Lie group, that we discuss in Proposition 2.4.An application of this criterion (Example 2.6) provides an alternative and moreconceptual proof of [25, Thm 3.1].2.
Bundles of Lie groups
As mentioned in the Introduction, we are interested in studying problems in-variant under a family of Lie groups, rather than a fixed Lie group. Such conceptappears in the literature with slight variations, see [5, 10, 14, 16, 23, 26]. In orderto give a precise definition, recall that a
Lie groupoid G over a (connected) smoothmanifold Λ is a smooth manifold equipped with the following data: • two smooth submersions s , t : G →
Λ, called source and target maps; • an associative composition operation G × Λ G ∋ ( g, h ) → g · h ∈ G , whichis a smooth map that associates to each pair ( g, h ) with s ( g ) = t ( h ) thecomposition g · h ; • an involution i : G → G , which is a smooth map that associates to each g ∈ G its composition inverse i ( g ) = g − ; • a smooth map s I : Λ → G , that to each λ ∈ Λ associates to the unit s I ( λ ) =1 λ for composition. Definition 2.1.
A Lie groupoid G over Λ for which the source and target mapscoincide, i.e., s = t , is called a (smooth) bundle of Lie groups over Λ, or a (smooth)family of Lie groups parameterized by
Λ. In this case, we also write G = { G λ : λ ∈ Λ } , where G λ := s − ( λ ) the Lie group consisting of the inverse image of λ by s .As a first example, consider the trivial of bundle of Lie groups G = Λ × G , where G is a Lie group and s = t is the projection onto the first factor. In this case, allgroups G λ are isomorphic to G . Due to a result of Weinstein [26], see also Moerdijk[16, Sec 1.3 (c)], the same happens for possibly nontrivial bundles of Lie groups,provided all G λ are compact. An abstract groupoid is a category in which every arrow is invertible. A point g ∈ G with s ( g ) = x and t ( g ) = y should be, categorically, thought of as an arrow from x to y . EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 5
Proposition 2.2.
A bundle of Lie groups G over Λ such that s : G → Λ is a propermap must be locally trivial , i.e., every λ ∈ Λ has an open neighborhood U in Λ ,such that s − ( U ) = U × G λ . In particular, bundles of compact Lie groups arelocally trivial.
Consequently, since Λ is assumed connected, the groups G λ of a bundle of com-pact Lie groups are pairwise isomorphic. From the viewpoint of varying families ofsymmetry groups that we adopt in our applications, this still is a trivial situation.Namely, the symmetry group remains unchanged (up to isomorphism). Therefore,actual nontrivial cases can only occur when some of the groups G λ are noncompact.In this case, the different groups G λ need not be isomorphic, not even homotopyequivalent (see Example 2.6). Remark . Similar definitions for bundles of Lie groups can be given in other reg-ularity contexts. For example, this concept was considered in the analytic categoryin [23]. The main result of [23] was later proved also in the smooth category [5].For our applications, it will be useful to also have a notion of continuous bundlesof Lie groups (as opposed to the above smooth notion), see Definition 4.1.2.1.
Integrating bundles of Lie algebras.
In order to produce nontrivial exam-ples of bundles of Lie groups, we consider the problem of obtaining such bundlesfrom bundles of Lie algebras, in a fashion similar to Lie’s Third Theorem (but fora 1-parameter family of Lie algebras). Given a bundle of Lie groups G , denoteby Ver( G ) the vertical bundle of G , i.e., the vector bundle over G whose fiber over g ∈ G is ker d π ( g ) ⊂ T g G , which is the tangent space at g to G s ( g ) = s − ( s ( g )).The pull-back of this vector bundle by the identity section,(2.1) Lie( G ) := s ∗ I (cid:0) Ver( G ) (cid:1) , is a vector bundle over Λ, called the bundle of Lie algebras associated to the bundleof Lie groups G . The fiber Lie λ ( G ) of (2.1) over λ ∈ Λ is the Lie algebra of theLie group G λ . In the general theory of Lie groupoids, this construction is a specialcase of the Lie algebroid associated to a Lie groupoid , see [17, Sec 6.1].An abstract Lie algebroid (see [17, Sec 6.2] for definitions) is integrable if it isisomorphic to the Lie algebroid associated to a Lie groupoid. There exist (finite-dimensional) Lie algebroids that are not integrable [17, Sec 6.4], and finding gen-eral integrability criterions for Lie algebroids is a notoriously difficult problem, see[6, 18, 19]. Nevertheless, in the simpler context of bundles of Lie algebras, Douadyand Lazard [10] proved that any (finite-dimensional) bundle of Lie algebras can beintegrated to a bundle of Lie groups (which may be not locally trivial nor Haus-dorff), cf. [17, Ex 6.3 (2)]. We now present a simple proof of a particular caseof this result (in which the bundle obtained is Haussdorff, but possibly not locallytrivial). This particular case comes from integrating bundles of Lie subalgebras ofa given Lie algebra b g , cf. [19]. Proposition 2.4.
Let Λ be a manifold and b G be a Lie group with Lie algebra b g .Let g be a smooth subbundle of the trivial vector bundle Λ × b g over Λ , such that,for all λ ∈ Λ , the fiber g λ is a Lie subalgebra of b g . For all λ ∈ Λ , let G λ be theconnected subgroup of b G whose tangent space at is g λ , and set G := [ λ ∈ Λ ( { λ } × G λ ) ⊂ Λ × b G. R. G. BETTIOL, P. PICCIONE, AND G. SICILIANO
Assume that for all λ ∈ Λ , there exists a submanifold A ⊂ b G and an open neigh-borhood V of λ in Λ such that: (a) 1 ∈ A ; (b) T A ⊕ g λ = b g ; (c) for all λ ∈ V , G λ ∩ A = { } .Then, G is a bundle of Lie groups over Λ , whose associated bundle of Lie algebrasis Lie( G ) = g .Proof. It suffices to prove that G is a submanifold of Λ × b G . Define a smooth map f : g × A → Λ × b G, f ( λ, X, a ) = (cid:0) λ, a · exp( X ) (cid:1) . For all λ ∈ Λ, using (b), it is easily seen that d f ( λ , ,
1) : T ( λ , g ⊕ T A → T λ Λ ⊕ b g is an isomorphism. Thus, by the inverse function theorem (up to reducing the sizeof A ), f restricts to a diffeomorphism from the product W × A to an open subset B = f ( W × A ) of Λ × b G , where W is a neighborhood of ( λ ,
0) in g . We claim thatthe preimage f − ( H ∩ B ) is given by W × { } . Namely, (cid:0) λ, a · exp( X ) (cid:1) ∈ G if andonly if a · exp( X ) ∈ G λ , and since exp( X ) ∈ G , this occurs if and only if a ∈ G λ . Byassumption (c), this implies a = 1, proving our claim. It follows that G is a smoothsubmanifold of Λ × b G near all points of the form ( λ , G at the other points, let us use thefollowing fact, proved in Lemma 2.5 below. For all λ ∈ Λ and all h ∈ G λ , thereexists a smooth function s : U → b G defined in an open neighborhood U of λ , suchthat s ( λ ) = h and s ( y ) ∈ G y for all y ∈ U . Fixed an arbitrary λ and an arbitrary h ∈ G λ , let s : U → b G be any such function, and consider the diffeomorphism φ s : U × b G → U × b G, φ s ( y, g ) = (cid:0) y, s ( y ) · g (cid:1) . Such a map carries a neighborhood of ( λ,
1) to a neighborhood of ( λ, h ). Moreover,it preserves
G ∩ ( U × b G ), because s ( y ) ∈ G y for all y ∈ U . Since sufficiently smallneighborhoods in G of all points of the form ( λ,
1) are submanifolds of Λ × b G , and h is arbitrary, it follows that G is a submanifold of Λ × b G . (cid:3) We have used the following existence result of smooth local sections of G , withprescribed value at some given point: Lemma 2.5.
In the notations above, for all λ ∈ Λ and h ∈ G λ , there exists anopen neighborhood U of λ in Λ and a smooth function s : U → b G such that s ( λ ) = h and s ( y ) ∈ G y for all y ∈ U .Proof. Since G λ is connected, there exist a finite sequence X , . . . X n in g λ suchthat h = exp( X ) · . . . · exp( X n ), since every neighborhood of 1 generates G λ . Then,one can find smooth extensions u , . . . , u n of the X i ’s to smooth local sections ofthe vector bundle g . The map s ( y ) = exp (cid:0) u ( y ) (cid:1) · . . . · exp (cid:0) u n ( y ) (cid:1) is the desiredsmooth local section of G . (cid:3) The tangent space at the point ( λ ,
0) of g is canonically identified with the horizontal plusvertical direct sum T λ Λ ⊕ g λ . For v ∈ T λ Λ, Z ∈ g λ and w ∈ T A , d f ( λ , , v, Z, w ) =( v, Z + w ), and this map is an isomorphism because of (b). φ s is clearly smooth, and its inverse is φ − s ( y, g ) = (cid:0) y, s ( y ) − g (cid:1) , which is also smooth. EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 7
We now discuss an example to which the above result applies, concerning thebundle of Lie groups formed by the (identity connected component of) isometrygroups of simply-connected space forms of constant curvature λ ∈ R . Example 2.6.
Given n ≥
2, set b g = gl ( n + 1) and b G = GL ( n + 1). For each λ ∈ R ,consider the injective linear map(2.2) L λ : so ( n ) × R n → b g , L λ ( D, u ) = (cid:18) − λu t u D (cid:19) , where D ∈ so ( n ) (i.e., D is an anti-symmetric matrix), u is a column vector and u t is the transpose of u . The map R ∋ λ L λ ∈ Hom (cid:0) so ( n ) × R n , b g (cid:1) is smooth, thusthe images of L λ form a smooth family of subspaces of b g . For all λ ∈ R , the imageof L λ is a Lie subalgebra g λ of b g with dimension n ( n +1), and g = S λ ∈ R ( { λ }× g λ )is a smooth vector subbundle of the trivial bundle R × b g .For λ = 0, it is easy to see that g λ is the Lie algebra of the Lie group SO ( η λ ) ⊂ b G of isomorphisms of R n +1 that preserve the nondegenerate symmetric bilinear form η λ , whose matrix in the canonical basis of R n +1 is: η λ ∼ = √ λ √ λ I n ! for λ > , and η λ ∼ = (cid:18) − √− λ √− λ I n (cid:19) for λ < , where I k denotes the k × k identity matrix. Set G λ := SO ( η λ ) if λ = 0, and G := (cid:18) R n SO ( n ) (cid:19) ⊂ b G . Then, we have the following isomorphisms with theisometry groups of space forms: G λ ∼ = SO ( n + 1) , (identity component of) isometry group of S n , if λ > , R n ⋊ SO ( n ) , (identity component of) isometry group of R n , if λ = 0 , SO ( n, , (identity component of) isometry group of H n , if λ < . We claim that(2.3) G := [ λ ∈ R ( { λ } × G λ ) ⊂ R × b G is a bundle of Lie groups over R . For all λ ∈ R , the subspace a ⊂ b g given by a = (cid:18) R R n n (cid:19) is a complement of g λ in b g , where Sym n denotes the space of n × n symmetric matrices. Denote by A the submanifold of b G consisting of matricesof the form (cid:18) a v t B (cid:19) , where a is a positive real number, v is an arbitrary vectorin R n and B is a positive symmetric matrix. Clearly, 1 ∈ A and T A = a . It is astraightforward computation that A ∩ G λ = { } for all λ ∈ R . Thus, the aboveclaim follows from Proposition 2.4. For λ = 0, G λ consists of matrices of the form (cid:18) a v t w B (cid:19) , where a ∈ R , v, w ∈ R n , B ∈ gl ( n ), av + λB t w = 0, a + λ k w k = 1, vv t + λB t B = λ I n . Given (cid:18) a v t w B (cid:19) ∈ G λ ∩ A , one sees easilythat it must be w = 0, v = 0, a = 1 and B = I n . Since B is (symmetric and) positive, theidentity B = I n implies B = I n . Thus, A ∩ G λ = I n +1 for λ = 0. Similarly, A ∩ G = I n +1 . R. G. BETTIOL, P. PICCIONE, AND G. SICILIANO
Remark . Note that the above bundle of Lie groups (2.3) is not locally trivialat 0 ∈ R . This is clear since G λ is noncompact for λ ≤ λ > G λ and G λ are not even homotopy equivalent when λ ≤ < λ . Remark . A totally analogous construction as in Example 2.6 can be done for theisometry group of symmetric spaces, as in [24]. Let ( M, g ) be a compact symmetricspace, and choose p ∈ M . Denote by g the Lie algebra of the isometry groupof ( M, g ) and by k the Lie algebra of the stabilizer of p . Consider the reductivedecomposition g = k ⊕ m , with [ k , k ] ⊂ k , [ k , m ] ⊂ m and [ m , m ] ⊂ k . Then, for λ ∈ R ,one considers a deformation [ · , · ] λ of the Lie bracket [ · , · ] of g , by setting:[ x, y ] λ = [ x, y ] , [ x, v ] λ = [ x, v ] , [ v, w ] λ = λ · [ v, w ] , for all x, y ∈ k , v, w ∈ m . Clearly, [ · , · ] λ is equal to [ · , · ] when λ = 1, while for λ = − M, g ). Denote by g λ the Lie algebra g endowed with the Lie bracket [ · , · ] λ . Then, as in Example 2.6, g λ is isomorphic to g , m ⋊ k or g − , according to the cases λ > λ = 0 and λ < The regular case
Let X be a Banach manifold, f : X → R a sufficiently regular function and x ∈ X a critical point of f . It is generally not a convenient assumption that thesecond derivative d f ( x ) : T x X → T x X ∗ is a Fredholm map. Namely, most Ba-nach spaces do not admit any Fredholm operator to their dual. On the other hand,in the cases of interest for applications to geometric variational problems, secondderivatives are represented by differential operators that are
Fredholm when consid-ered between suitable spaces . For instance, strongly elliptic self-adjoint differentialoperators of order d , defined on the space of sections of some vector bundle E , areFredholm operators of index 0 from Γ k,α ( E ) to Γ k − d,α ( E ), for every d ≤ k andevery α ∈ ]0 , k,α ( E ) denotes the space of sections of E of class C k,α , seefor instance [28]. A natural abstract formulation of such Fredholmness property isgiven as follows. Definition 3.1.
Given a Banach manifold X and a C k +1 -function f : X → R , k ≥
1, an auxiliary Fredholm structure for (
X, f ) consists of: • a smooth Banach bundle E on X , with continuous inclusion T X ⊂ E ; • a (possibly non-complete) inner product h· , ·i x on each fiber E x , for all x ∈ X ,satisfying the following properties: • there exists a C k -section δf : X → E such that d f ( x ) = h δf ( x ) , ·i x for all x ∈ X (in particular, d f ( x ) = 0 if and only if δf ( x ) = 0); • for all critical points x ∈ X , there exists a Fredholm operator J x : T x X →E x of index 0, such that(3.1) Hess f ( x )[ v, w ] = h J x ( v ) , w i x , for all v, w ∈ T x X, where Hess f ( x ) is the bounded symmetric bilinear form on T x X given bythe Hessian of f at x .Note that ker (cid:0) Hess f ( x ) (cid:1) = ker( J x ), and that the map δf is a gradient-like map for f . From (3.1), J x is symmetric with respect to h· , ·i x . Nevertheless,this symmetry alone does not imply in general that J x has Fredholm index 0, see EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 9
Appendix A for a thorough discussion. In typical applications, δf is a quasi-linearelliptic differential operator between suitable spaces and J x is its linearization, i.e.,a linear elliptic differential operator, for which the above assumptions are satisfied. Remark . For all x ∈ X , denote by 0 x the zero of the fiber E x , and by π ver ( x ) : T x E → E x the vertical projection. Given a critical point x of f , i.e., δf ( x ) = 0 x , the derivative d( δf )( x ) is a linear map from T x X to T x E . Theoperator J x above can be defined as the vertical derivative of δf at x , given by:(3.2) π ver ( x ) ◦ (cid:0) d( δf )( x ) (cid:1) : T x X −→ E x . Finally, let us give a fiber bundle version of a result in [2] on the intersectionof Banach submanifolds. Aiming at applications to not necessarily smooth groupactions, we only assume differentiability along one fiber.
Proposition 3.3.
Let M be the total space of a fiber bundle with base A . For all a ∈ A , denote by M a the fiber over a , which is assumed to be a finite-dimensionalmanifold. Let N be a (possibly infinite-dimensional) Banach manifold, and Q ⊂ N a Banach submanifold. Assume that χ : M → N is a continuous function such thatthere exists a ∈ A and p ∈ M a with: (a) χ ( p ) ∈ Q ; (b) χ a = χ (cid:12)(cid:12) M a : M a → N is of class C ; (c) d χ a (cid:0) T p M a (cid:1) + T χ ( p ) Q = T χ ( p ) N .Then, for a ∈ A near a , χ ( M a ) ∩ Q = ∅ .Proof. Given the local character of the result, we can use a trivialization of the fiberbundle M around a , and assume that M = A × M , where M is a finite-dimensionalmanifold diffeomorphic to the fibers. For this case, the proof of the result is givenin [2, Prop 3.4], using a topological degree argument. Note that transversalityarguments cannot be used here, since we are not assuming differentiability of χ along all the fibers in a neighborhood of a . (cid:3) Given Banach spaces E and F , denote by Hom( E, F ) the space of boundedlinear operators from E to F , and for T j ∈ Hom( E j , F ), define ( T ⊕ T )( e , e ) := T ( e ) + T ( e ). Lemma 3.4.
Let E , E and F be Banach spaces. The following is open in Hom( E , F ) × Hom( E , F ) : (cid:8) ( T , T ) ∈ Hom( E , F ) × Hom( E , F ) : ( T ⊕ T ) : E ⊕ E → F is surjective and has complemented kernel (cid:9) . Proof.
Set E = E ⊕ E ; it is well-known that the set of T ∈ Hom(
E, F ) thatare surjective and have complemented kernel is open in Hom(
E, F ). Moreover, themap Hom( E , F ) × Hom( E , F ) ∋ ( T , T ) T ⊕ T ∈ Hom(
E, F ) is linear andcontinuous. (cid:3)
We now give the proof of the Theorem in the Introduction, when all the aboveregularity hypotheses are fulfilled.
Proof (Regular case).
Recall that we are denoting by D λx the image of the linearmap d β λx (1) : g λ → T x X . Our assumptions on the regularity of the action implythat d β λx (1) depends continuously on ( x, λ ). We claim that we can find a Banachsubmanifold S ⊂ X through x , satisfying the following properties: (1) T x S ⊕ D λ x = T x X ;(2) T x S + D λx = T x X for all x ∈ S and all λ in a neighborhood U ⊂ Λ of λ ;(3) there exists a neighborhood e V ⊂ X of x such that G λ · S ⊃ e V for all λ ∈ U .The existence of such S can be argued as follows. First, observe that D λ x has finitedimension, thus it is complemented in T x X ; let S be any closed complement of D λ x in T x X , and let S ⊂ X be any smooth submanifold through x with T x S = S .Clearly, for such S , equality (1) holds.Moreover, we claim that the set of ( λ, x ) ∈ Λ × S for which T x S + D λx = T x X isopen, which implies easily that, by possibly taking a smaller S , also property (2)holds. To prove the claim, use Lemma 3.4 applied to the following pair of operators: T = T ( x, λ ) = d β λx (1) : g λ → T x X , and T = T ( x ) : T x S → T x X the inclusion of T x S into T x X . By using suitable trivializations of the bundles g = S λ ( { λ }× g λ ) and T X near λ and x , clearly one can assume that the domains and the range of theseoperators are fixed Banach spaces. Observe that T x S + D λx = T x X is equivalent to T ( x, λ ) + T ( x ) being surjective. On the other hand, the kernel of T ( x, λ ) + T ( x )is always complemented, because it is finite-dimensional. Namely, T ( x ) is injective,and the domain of T ( x, λ ) is finite-dimensional. Hence, Lemma 3.4 implies thatthe set of ( λ, x ) ∈ Λ × S for which T x S + D λx = T x X is open.Property (3) holds if we show that β λx ( G λ ) ∩ S = ∅ , for ( λ, x ) sufficiently closeto ( λ , x ). In the regular case that is being considered here, this follows easily bya transversality argument. Namely, by the continuity of d β λx , property (1) impliesthat the map β λx is transversal to S for ( λ, x ) sufficiently close to ( λ , x ).Once the existence of a submanifold S for which the above properties hold hasbeen established, the proof of our result is obtained from the following argument.Property (2) implies that, for λ ∈ U , the critical points of the restriction f λ (cid:12)(cid:12) S : S → R are in fact critical points of f λ . By property (3), for all λ ∈ U and all y ∈ e V ,d f λ ( y ) = 0 if and only if there exists z ∈ S , with y ∈ G λ · x , such that d( f λ | S )( z ) = 0.Thus, the equivariant implicit function theorem is reduced to the standard implicitfunctions theorem for the restriction f λ | S .From here, the rest of the proof goes along the same line as the proof of theequivariant implicit function theorem with fixed group of symmetries, as presentedin [2]. This part of the proof, that will be omitted here, uses property (1) of S , thegradient-like maps δf λ , the vertical derivative of δf λ at x , and the equivariantnondegeneracy assumption. (cid:3) The non-regular case
In several interesting situations arising from geometric variational problems, theexistence of an implicit function can only be obtained in a framework without theregularity properties assumed in Section 3. We use the example of the constantmean curvature (CMC) variational problem to address some of those regularityissues, and then proceed to a case-by-case discussion (see Subsections 4.1, 4.2, 4.3)of how to weaken various hypotheses of the result proved in Section 3.In the CMC variational problem, one searches for minimizers of the area func-tional subject to a volume constraint, in the space of codimension 1 unparametrizedembeddings (or immersions) of a compact manifold M in a Riemannian manifold i.e., embeddings modulo reparametrizations. EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 11 M , see Subsection 5.1 for details. Equivariance in this scenario comes from theleft-composition action of the isometry group of M on the space of embeddings of M into M . There are several technical issues concerning this problem: • the space X of C k,α unparametrized embeddings of M into M is not adifferentiable manifold. It only admits a natural atlas of local charts, whichare not differentiably compatible; • although the isometry group of M is a Lie group, its action on X is onlycontinuous, and differentiable only on a dense subset (consisting of smooth,i.e., C ∞ , embeddings); • due to topological reasons, it may be convenient to not define the groupaction globally on X , but rather describe it as a local action .A detailed discussion of the first two points above can be found in [1]. The questionof locality for the group action is basically a matter of technicalities, and dealingwith this situation entails no essential modification of the theory presented here.The interested reader may find a discussion of local actions in the context of ourequivariant implicit function theory in [2]. Let us simply emphasize here that animportant consequence of extension to local actions is the corresponding weakeningon the assumption that X admits a global differentiable structure. Namely, if G isa group that acts on X , then one has a local action of G on any open subset of X .In particular, for the validity of the equivariant implicit function theorem, it willbe enough to have that X admits an atlas of local charts that are only continuouslycompatible, i.e., an atlas defining a global topological manifold structure on X ,provided that the relevant functions be differentiable in each local chart.This seemingly unusual situation actually occurs quite often in problems in-variant under reparametrizations. In the case at hand, the space X of C k,α un-parametrized embeddings of M into M admits a local parametrization near x ∈ X in terms of sections of the normal bundle of x ; and it is well-known that two suchlocal parametrizations are only continuously (and not differentiably) compatible.Nevertheless, smooth action functionals defined on the space of embeddings, thatare invariant under diffeomorphisms of the source manifold, define continuous func-tions on the quotient space of unparametrized embeddings, that are also smooth inevery local chart. This property is discussed thoroughly in [1].4.1. Non-regular group actions.
An important issue is the lack of differentia-bility of the maps β λx ( g ) = g · x , for all x . In typical geometric applications, x isa given map of class C k,α and β λx is differentiable (at 1 λ ) only when x has higherregularity, say C ∞ . The set of such x is in general dense, and the corresponding(densely defined) map ( x, λ ) d β λx (1 λ ) admits a continuous extension to all x , asa section of an appropriate bundle. This will be discussed in concrete examples inthe sequel; here we give abstract axioms to deal with this situation.The regularity assumption for the action of G on X in the equivariant implicitfunction theorem can be replaced with the assumption that, for all λ ∈ Λ, the map β λx is differentiable at 1 λ for all x belonging to a dense subset X ′ ⊂ X containing x , and assuming the existence of a C k -vector bundle Y over X , together with C k -bundle morphisms j : E → Y ∗ and κ : T X → Y , satisfying the following properties: • κ is injective; • κ ∗ ◦ j : E →
T X ∗ coincides with the inclusion E in T X ∗ via the innerproduct h· , ·i (from which it follows that also j must be injective); • the map Λ × X ′ ∋ ( λ, x ) κ x ◦ d β λx (1 λ ) ∈ Hom( g λ , Y x ) has a continuousextension to a section of the vector bundle Hom(Lie( G ) , Y ) over Λ × X .With the above setup, one proves the existence of a submanifold S of X through x , such that T x S is a complement to D λ x , and such that the standard implicitfunction theorem holds for the restriction of f λ to S around ( x , λ ). The proof ofthis fact is totally analogous to the proof of the same result in the case of fixed groupof symmetries discussed in [2], and will be omitted here. The argument is concludedby showing the existence of a neighborhood e V ⊂ X of x such that G λ · S ⊃ e V for all λ ∈ U ; for this, the transversality argument used in the regular case cannotbe used here. Instead, the existence of such e V is obtained from Proposition 3.3,applied to the following setup: A = Λ × X , M is the fiber bundle over Λ × X whose fiber over ( λ, x ) is G λ , a = ( λ , x ), p = (cid:0) ( λ , x ) , λ (cid:1) , N = X , Q = S ,and χ (cid:0) ( λ, x ) , g (cid:1) = β λx ( g ) = g · x . Assumption (b) of Proposition 3.3 is satisfied,since by assumption x ∈ X ′ and therefore χ (cid:0) ( λ , x ) , · (cid:1) = β λ x is differentiable.Assumption (c) holds because T x S is a complement to D λ x .4.2. Actions by homeomorphisms.
In the proof of our equivariant implicit func-tion theorem, we have used the assumption that the group actions on X are bydiffeomorphisms in order to guarantee that, given λ ∈ Λ and a critical point x of f λ , then the G λ -orbit of x consists of critical points of f λ . Namely, the proof ofthis fact involves the derivative of the map x g · x , which in principle does notexist if the action of G λ on X is assumed to be only by homeomorphisms. Thus,when dealing with actions by homeomorphisms, an explicit invariance property forcritical points has to be assumed: for all λ , the set (cid:8) x ∈ X : d f λ ( x ) = 0 (cid:9) is G λ -invariant. Having such assumption satisfied, the equivariant implicit functiontheorem holds also in the case of actions by homeomorphisms.In practical terms, one obtains invariance of the set of critical points when the G λ -action on X lifts to an action on the vector bundle E introduced in Section 3,by linear isomorphisms on the fibers, and the gradient-like map δf λ : X → E isequivariant with respect to such action. In such a situation, the null section of E isfixed by the G λ -action. Thus, the set of critical points of f λ , which coincides withthe set of zeroes of the gradient-like map δf λ , is G λ -invariant.4.3. Actions of continuous bundles of Lie groups.
While the existence of aLie group structure on each symmetry group G λ is a central point in our theory,it may be interesting to weaken the regularity assumption on the dependence onthe parameter λ . More precisely, one can consider continuous , rather than smooth,bundles of Lie groups (cf. Definition 2.1). Definition 4.1.
Let Λ be a topological space, and, for all λ ∈ Λ, let G λ be a Liegroup. The set G = S λ ∈ Λ ( { λ } × G λ ) is a continuous bundle of Lie groups over Λif the following properties are satisfied: • G is a groupoid with a topology whose restriction to each fiber { λ } × G λ coincides with the topology of G λ ; • the composition operation G × Λ G ∋ ( g, h ) → g · h ∈ G , is continuous; • the involution i : G ∋ g g − ∈ G is continuous; i.e., there is a left-action of G λ on E , such that the projection E → X is equivariant withrespect to such action. EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 13 • there exist local trivializations of G , in the following sense: for all λ ∈ Λ,there exists a neighborhood V of λ in Λ, a neighborhood U of 1 λ in G λ , aneighborhood Z of ( λ , λ ) in G and a homeomorphism h : V × U → Z suchthat h ( λ , g ) = ( λ , g ) for all g ∈ U and for which the following diagramcommutes:(4.1) V × U h / / π ●●●●●●● Z π (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦ V where π : V × U → V is the projection onto the first variable, and π : G →
Λis the natural projection; • commutativity of (4.1) means that h carries fibers { λ } × U ⊂ V × G λ ontofibers G λ ∩ Z ⊂ G . It is required that the restriction of h to each fiberbe differentiable, and that the fiber derivative ∂ h : U × T G λ → T G be acontinuous map. Remark . It is not hard to prove that a smooth bundle of Lie groups, as inDefinition 2.1, is also a continuous bundle of Lie groups. The existence of localtrivializations for a smooth bundle of Lie groups is obtained easily using standardresults regarding the local form of submersions, applied to the source map s : G →
Λ.The above axioms imply that the identity section s I : Λ → G , s I ( λ ) = 1 λ for all λ ∈ Λ, is continuous. Namely, for the continuity at any fixed λ , consider a localtrivialization h : V × U → Z as above, and set g λ = h ( λ, λ ) for all λ ∈ V . Then, s I ( λ ) = g − λ · h ( λ, λ ), which is continuous by the above axioms.It also follows easily that the set Lie( G ) := S λ ∈ Λ ( { λ }× g λ ) is a topological vectorbundle over Λ. Namely, given a local trivialization h : V × U → Z as in (4.1), thenthe map e h : V × U → e Z defined by e h ( λ, g ) = h ( λ, λ ) − h ( λ, g ) is another localtrivialization that satisfies e h ( λ, λ ) = 1 λ for all λ ∈ V . Then, vector bundle localtrivializations for Lie( G ) are obtained by differentiating at 1 λ such a map e h .The continuity of the action of G on X is intended in the sense that the map β : G × X → X , defined by β ( λ, g, x ) = β λx ( g ) = g · x , is continuous in all threevariables. The minimal regularity requirements for the action are that • there exists a dense subset X ′ ⊂ X such that β ( λ, · , x ) is differentiable at1 λ for all λ ∈ Λ and all x ∈ X ′ ; • as in Subsection 4.1, given vector bundles E and Y , with morphisms j : E →Y ∗ and κ : T X → Y , the map k x ◦ d β ( λ, λ , x ) must admit a continuousextension to a global section of the bundle Hom(Lie( G ) , Y ).Equipped with the above topological framework, the proof of the equivariant im-plicit function theorem in Section 3 carries over verbatim to the case of continuousbundles of Lie groups.5. Applications to Geometric Variational Problems
In this section, we discuss various applications of the above abstract deformationresults to geometric variational problems. The necessity of the non-regular exten-sions is illustrated in two of the three applications discussed, namely the variationalproblems of finding CMC and Hamiltonian stationary Lagrangian submanifolds. Inthese two problems, invariance under reparametrizations causes the group action to be not differentiable at every point. The third application, concerning harmonicmaps between manifolds, gives an example of the regular case, where these exten-sions are not needed.5.1.
Deformation of CMC hypersurfaces.
Let M be a compact manifold, withdim M = m , and let M be a differentiable manifold, with dim M = m + 1, endowedwith a Riemannian metric g . For α ∈ ]0 , ,α ( M, M ) the set of allembeddings of class C ,α of M into M , which is an open subset of the Banach space C ,α ( M, M ) of all C ,α -functions from M to M . Two embeddings x i : M → M , i = 1 , , are said to be isometrically congruent if there exists a diffeomorphism φ of M and an isometry F of M such that x = F ◦ x ◦ φ . Recall that the meancurvature H x of an embedding x : M → M is the norm of the mean curvature vector ~H x of x , which is the trace of the second fundamental form S x . The embedding issaid to have constant mean curvature H (in short, CMC), if H x = H .The CMC embedding problem has a convenient variational framework that wenow describe. Suppose x : M → M is a CMC embedding such that x ( M ) is theboundary of a bounded domain of M . Let X be the (component of x of the) setof C ,α -unparametrized embeddings of M into M , i.e., the quotient of (componentcontaining x of) the space of C ,α -embeddings of M into M by the action of thediffeomorphism group of M . In other words, X is the set of C ,α -submanifolds of M that are diffeomorphic to M . Such a space admits a global topological manifoldstructure, modeled on the Banach space C ,α ( M ) of real-valued functions on M of class C ,α . Given a smooth embedding x : M → M , denote by [ x ] ∈ X thecorresponding class. Unparametrized embeddings in a neighborhood of [ x ] ∈ X areparametrized by C ,α -sections V of the normal bundle of x near the null section.Given one such V , the associated unparametrized embedding is the class of theembedding M ∋ p exp x ( p ) V p ∈ M , where exp is the exponential map of ( M , g ).Since x ( M ) is the boundary of a domain in M , the normal bundle of x ( M ) isorientable. Thus, its sections are of the form V = h ~n x , where ~n x is the unitnormal field of x , and such section V is identified with the function h : M → R . Inparticular, for all x ∈ X , the tangent space T [ x ] X is identified with C ,α ( M ).Let g λ be a smooth family of metrics on M , parametrized by λ ∈ Λ, where Λ isa smooth manifold. Given an unparametrized embedding [ x ], where the image of x is the boundary of a bounded domain Ω x ⊂ M , denote by Area λ ( x ) the volume of( M, g λ ), where g λ is the pull-back metric x ∗ ( g λ ). Denote by Vol λ ( x ) the volumeof Ω x relatively to the volume form of g λ . For a given H ∈ R , set(5.1) f : X × Λ → R , f ([ x ] , λ ) = Area λ ( x ) + H Vol λ ( x ) . Regularity properties of this functional in the space of unparametrized embeddingsare discussed in [1]. If x is a smooth embedding, up to using a local chart around[ x ] ∈ X as an identification, we can consider the first variation of the above func-tional with respect to a variation V = h ~n x , which is well-known to be(5.2) ∂ f ([ x ] , λ ) V = Z M ( H − H x ) h vol g λ , where H x is the mean curvature function of the embedding x : M → M . As aresult, critical points of f λ : X → R are precisely the classes of smooth embeddingsof M into M that have constant mean curvature equal to H , when M is endowed EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 15 with the metric g λ . The corresponding Jacobi operator J x of a CMC embedding x is the linear elliptic differential operator(5.3) J x ( h ) = ∆ x h − (cid:0) Ric( ~n x ) + kS x k (cid:1) h, where ∆ x is the (positive) Laplacian of functions on M relative to the metric x ∗ ( g λ ), Ric( ~n x ) is the Ricci curvature of M evaluated on the unit normal ~n x and S x is the second fundamental form of x . A Jacobi field along x is a smooth function h : M → R satisfying J x ( h ) = 0.Let G λ = Iso( M , g λ ) be the isometry group of ( M , g λ ), which is a finite-dimensional Lie group. For each λ ∈ Λ, the Lie group G λ acts on X by left-composition g · [ x ] = [ g ◦ x ]. This action is continuous , but not differentiable on allof X . Both Area λ and Vol λ on (5.1) are G λ -invariant functions on X , and henceso is f λ . This invariance implies that symmetries of the ambient, encoded in theform of Killing fields, induce certain Jacobi fields on every CMC embedding. Definition 5.1.
Given a Killing vector field K on ( M , g λ ), the function h K = g λ ( K, ~n x ) is a Jacobi field. Jacobi fields of this form are called Killing-Jacobifields , and h K is said to be induced by K . The CMC embedding x is said to be equivariantly nondegenerate if all the Jacobi fields along x are Killing-Jacobi fields.In other words, x is equivariantly nondegenerate if all infinitesimal CMC defor-mations of x with the same mean curvature arise from isometries of the ambientspace. Equivariant nondegeneracy of CMC embedding is the key property to obtainthe following deformation results for CMC embeddings: Theorem 5.2.
Let g λ be a family of Riemannian metrics on M , parametrized by λ ∈ Λ , where Λ is a smooth manifold. Let x be an equivariantly nondegenerateembedding of M into ( M , g λ ) with constant mean curvature H , such that x ( M ) is the boundary of a bounded domain of M . Assume that G λ = Iso( M , g λ ) , λ ∈ Λ , forms a continuous bundle of Lie groups G = { G λ : λ ∈ Λ } . Then, thereexists a neighborhood V of λ in Λ , and a smooth function x : V → Emb ,α ( M, M ) satisfying: (a) x ( λ ) is an embedding of M into ( M , g λ ) with constant mean curvature H for all λ ∈ V ; (b) x ( λ ) = x .Moreover, given λ ∈ V , any other embedding of constant mean curvature H of M into ( M , g λ ) sufficiently close to x ( λ ) in Emb ,α ( M, M ) is isometrically congruentto x ( λ ) .Proof. This is an application of our equivariant implicit function theorem to theabove variational setup that requires all the non-regularity extensions discussed inSection 4. As indicated above, X is the space of unparametrized embeddings of M into M of class C ,α , and f is given by (5.1). For all λ ∈ Λ, the Lie group G λ actson X by left-composition, and the maps β λ [ x ] : G λ → X are smooth when [ x ] is theclass of a smooth embedding. The set X ′ of such embeddings is dense in X . Inparticular, the image of d β λ [ x ] (1 λ ) coincides with the space of Killing-Jacobi fieldsalong x , so that the equivariant nondegeneracy assumption on x is precisely theequivariant nondegeneracy assumption of our implicit function theorem.All the other objects involved in the statement and the proof of the abstractimplicit function theorem, described in Section 3 and Section 4, are as follows: • E is the vector bundle with fiber over [ x ] the Banach space of C ,α -sectionsof the normal bundle of x . Recall that T [ x ] X is the Banach space of C ,α -sections of the normal bundle of x , and the inclusion T X ⊂ E is obvious. • The inner product h· , ·i [ x ] on E [ x ] is the L -paring h h , h i [ x ] = R M h h vol g ∗ with respect to the volume form of a background metric g ∗ ; • Y is the vector bundle on X , whose fiber at the point [ x ] is the Banachspace of C ,α -sections of the normal bundle of x . The bundle morphism κ : T X → Y is the obvious inclusion, and the morphism j : E → Y ∗ isinduced by the L -pairing above; • identifying the Lie algebra g λ with the space of (complete) Killing vectorfields on ( M , g λ ), for [ y ] ∈ X ′ , the map d β λ [ y ] (1 λ ) : g λ → T [ y ] X associatesto a Killing vector field K the g λ -orthogonal component of K along y ; • δf λ ([ x ]) = ( H − H x ) ξ g λ , where H x is the mean curvature function of x (which is a C ,α -function on M ) and ξ g λ is the positive function satisfying ξ g λ vol g ∗ = vol g λ , see (5.2); • the vertical derivative of δf λ at [ x ] is identified with the Jacobi operator J x , as indicated in (3.2). As a linear elliptic operator of second-order, thisis a Fredholm operator of index 0 from C ,α ( M ) to C ,α ( M ). (cid:3) More general results on CMC deformations.
In view to concrete appli-cations, it is worth discussing a few generalizations of Theorem 5.2 that allow toobtain deformation results for CMC immersions , and not only embeddings .5.2.1.
Generalized volume functionals.
Let us start with some remarks on the as-sumption that x ( M ) should be the boundary of a domain in M . First, it mustbe observed that Theorem 5.2 does not hold in full generality if such assumptionis dropped, see [2, Ex 4.11]. In particular, with the above statement, the resultof Theorem 5.2 does not apply to CMC embeddings of manifolds with boundary.There exist, however, interesting generalizations of the result obtained by weak-ening this assumption. The crucial observation is that such assumption is neededfor the variational characterization of CMC embeddings, which uses the volumefunctional Vol λ . For the validity of the result, one may assume more generally theexistence of an invariant volume functional in a neighborhood of x in the space ofembeddings of M into M . Details of the definition of invariant volume functionalsare discussed in [2, App B]. Let us simply list some conditions under which invariantvolume functionals exist: • M noncompact and Iso( M , g ) compact; • ∂M = ∅ and M noncompact; • M noncompact with H m ( M , R ) = 0, or, more generally, if the image of x is contained in an open subset of M whose m th de Rham cohomologyvanishes.The last item includes, for instance, the cases M = R m +1 and S m +1 . Moreover,manifolds of the form M m +1 = R k × N m +1 − k , k ≥
1, have trivial m th de Rhamcohomology; and manifolds of the form M m +1 = S k × N m +1 − k , k ≥
1, have opendense subsets with trivial m th de Rham cohomology. The Riemannian metric g ∗ is a (fixed) reference metric, an can be chosen arbitrarily. Itsimply serves the purpose of inducing an L -pairing that does not vary with λ . EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 17
Free immersions.
As we have pointed out, the space of unparameterizedembeddings of a manifold M into another manifold M is described as the space oforbits of the action of the diffeomorphism group of M acting by right-compositionin the space of embeddings of M into M . Such action is free, and this fact playsan important role in establishing the (local) smooth structure for the space ofunparameterized embeddings. A general immersion x : M → M can have non-trivial stabilizer in the group of diffeomorphisms; for instance, an iterated closedcurve in M is an immersion of S whose stabilizer is a finite cyclic group.The variational theory for parametrization invariant geometric functionals canbe extended to the case of immersions whose stabilizer is trivial, i.e., the so-called free immersions of M into M . More precisely, an immersion x : M → M is free if the only diffeomorphism ϕ : M → M such that x ◦ ϕ = x is the identity. Thedifferentiable structure of free immersions is studied in detailed, for the smoothcase, in [4]. The appropriate extensions to the C ,α -case, needed for dealing withthe CMC variational problem, can be carried out as in [1]. We point out that beingfree is a rather weak assumption; for instance, if there exists at least one point p such that the inverse image of x ( p ) consists of a single point, then the immersion x is free (see [4, Lemma 1.4]). Theorem 5.2 holds, mutatis mutandis , under the moregeneral assumptions that x : M → M is a free immersion. A general statementincluding this case will be given below (Theorem 5.4).5.2.3. A generalized equivariant nondegeneracy assumption.
As a motivation forthe introduction of a more general nondegeneracy assumption in Theorem 5.2, letus observe that great attention has been given recently to the study of submanifoldtheory (surfaces) in ambient spaces like M k = M ( k ) × R , where M ( k ) is a 2-dimensional space form of curvature k , see [9, 11, 12, 15, 20]. In this case, when k = 0, the isometry group of M k has dimension 4, while for k = 0, since M isthe standard Euclidean 3-space, the isometry group is 6-dimensional. Thus, if onewants to prove deformation results for surfaces in R to surfaces in M k , k = 0,Theorem 5.2 cannot be applied.Nonetheless, we observe that the isometry group of R contains a closed subgroup G such that, denoting G k = Iso( M k ), k = 0, the set (cid:8) ( k, G k ) : k ∈ R (cid:9) is a smoothbundle of Lie groups. Namely, it is enough to define G as the group of isometriesof R that preserve the splitting R × R . Clearly, such group is isomorphic to theproduct Iso( R ) × R , and has dimension 4. This motivates the following: Definition 5.3.
Let (
M , g ) be a Riemannian manifold, Iso( M , g ) the group ofisometries of ( M , g ), Iso ( M , g ) its Lie algebra, and let K be a Lie subalgebraof Iso ( M , g ). A codimension 1 CMC embedding x : M → M is said to be K -nondegenerate if every Jacobi field along x is a Killing-Jacobi field induced by someelement of K .In conclusion, we can formulate the following extension of Theorem 5.2: Theorem 5.4.
Let M m be a compact manifold (possibly with boundary) and let M m +1 be any differentiable manifold. Let g λ be a family of Riemannian metrics on M , parameterized by λ ∈ Λ , where Λ is a smooth manifold. Let λ ∈ Λ be fixed; for λ = λ , set G λ = Iso( M , g λ ) , and let G λ be a closed subgroup of Iso(
M , g λ ) , withLie algebra g λ . Let x be a free immersion of M into ( M , g λ ) having constantmean curvature H . Assume the following: • there exists an invariant volume functional in a neighborhood of x ( M ) ; • the family G λ , λ ∈ Λ , forms a continuous bundle of Lie groups; • x is g λ -nondegenerate.Then, there exists a neighborhood V of λ in Λ , and a smooth function x : V → C ,α ( M, M ) satisfying: (a) x ( λ ) is an immersion of M into ( M , g λ ) with constant mean curvature H ,and with x ( λ )( ∂M ) = x ( ∂M ) for all λ ∈ V ; (b) x ( λ ) = x .Moreover, given λ ∈ V , any other fixed boundary immersion of constant meancurvature H of M into ( M , g λ ) , sufficiently close to x ( λ ) in C ,α ( M, M ) is iso-metrically congruent to x ( λ ) . Examples where Theorem 5.4 is applied can be obtained by looking at CMCimmersions of manifolds with boundary conditions.
Example 5.5.
Let M be a compact manifold with boundary, with, say ∂M = N ∪ N , where N and N the two connected components of ∂M . Let x : M → R be a CMC immersion such that x ( N ) and x ( N ) belong to two parallel horizontalplanes, say x = a i , i = 1 ,
2. Here, we think of R with coordinates ( x , x , x ), so x = const. are the horizontal planes. We say that x is equivariantly nondegenerateif the only Jacobi fields J along x that are horizontal on ∂M are Killing-Jacobi fields.For k ∈ R , let M ( k ) be the 2-dimensional space form of curvature k . Note thatequivariant nondegeneracy does not imply that x is rotationally invariant.If x is equivariantly nondegenerate, then x can be smoothly deformed to CMCimmersions of M into the product M ( k ) × R , with the same constant mean cur-vature, and with boundary on the slices M ( k ) × { a i } , i = 1 ,
2. This follows fromTheorem 5.4, observing that any Killing vector field in R whose restriction to anontrivial closed horizontal curve is horizontal must be the sum of an infinitesimalhorizontal rotation and a horizontal translation. In other words, Theorem 5.4 isapplied to the Lie algebra g consisting of vector fields that generate isometries of R that preserve the splitting R × R . In particular, the result applies to everyequivariantly nondegenerate portion of nodoid or unduloid in R , with boundaryon horizontal planes.5.3. Deformation of harmonic maps.
Let ( M, g ) and ( M , g ) be Riemannianmanifolds. A C -map φ : M → M is said to be ( g , g )- harmonic if(5.4) ∆ g , g ( φ ) := tr( b ∇ d φ ) = 0where b ∇ is the connection on the vector bundle T M ∗ ⊗ φ ∗ ( T M ) over M inducedby the Levi-Civita connections ∇ of g and ∇ of g . When the source manifold M is compact, harmonic maps on M have the following variational characterization.Let X be the Banach manifold C ,α ( M, M ) of maps φ : M → M of class C ,α .Let Λ be a smooth manifold that parametrizes a family of pairs ( g λ , g λ ), where g λ More generally, any Killing vector field K in R which is horizontal at three noncollinearpoints lying in a horizontal plane, must be the sum of an infinitesimal horizontal rotation and ahorizontal translation. Write a general Killing field in R as K = D + W , where D ∈ so (3) isan infinitesimal rotation and W ∈ R is a translation. Assume P , P and P three noncollinearpoints on the plane z = z , and set v = −−−→ P P , v = −−−→ P P ; these are linearly independenthorizontal vectors. Then, ( Dv ) · e = ( Dv ) · e = 0, i.e., D is a horizontal infinitesimal rotation.From this, it also follows that W is horizontal. EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 19 is a metric on M and g λ is a metric on M , both of class C k for a fixed k ≥
3. Set(5.5) f : X × Λ → R , f ( φ, λ ) = Z M k d φ ( x ) k HS vol g λ , where vol g λ is the volume form (or density) of g λ and k d φ ( x ) k HS is the Hilbert-Schmidt norm of the linear map d φ ( x ). Note that the dependence on g λ in theabove formula is hidden in k d φ ( x ) k HS , which depends on the metrics of both sourceand target manifolds.For a given λ ∈ Λ, critical points of f λ ( φ ) = f ( φ, λ ) are precisely the ( g λ , g λ )-harmonic maps φ : M → M . For φ ∈ X , the tangent space T φ X is identified withthe Banach space C ,α ( φ ∗ T M ) of all vector fields along φ of class C ,α . Given asuch V ∈ T φ X , the first variation of f in this direction is given by:(5.6) ∂ f ( φ, λ ) V = Z M tr (cid:0) d φ ∗ ∇ V (cid:1) vol g λ = − Z M g λ (cid:0) ∆ g λ , g λ ( φ ) , V (cid:1) vol g λ , where the trace is meant on the entries d φ ∗ ( · ) ∇ ( · ) V . The correspondent Jacobioperator J along a ( g λ , g λ )-harmonic map φ is the linear differential operator:(5.7) J φ ( V ) = − ∆ V + tr (cid:0) R (d φ ( · ) , V )d φ ( · ) (cid:1) , defined in C ,α ( φ ∗ T M ). Here R is the curvature tensor of g λ , and ∆ V is a vectorfield along φ uniquely defined by(5.8) g λ (∆ V, W ) = div( ∇ V ∗ ) W − g λ ( ∇ V, ∇ W ) , W ∈ C ,α ( φ ∗ T M )i.e., ∆ V ( x ) = P i (cid:0) ∇ e i ∇ V (cid:1) e i , where ( e i ) i is an orthonormal basis of T x M . In thissituation, a vector field V along φ that satisfies J φ ( V ) = 0 is called a Jacobi field .There are two special types of Jacobi fields along a ( g λ , g λ )-harmonic map φ .Let G λ be the product of isometry groups Iso( M, g λ ) × Iso(
M , g λ ); there is anatural action of G λ on X given by(5.9) G λ × X ∋ (cid:0) ( g, g ) , φ (cid:1) g ◦ φ ◦ g − ∈ X. Clearly, (5.5) is invariant under this action. Using results from [21], one can provethat this action is smooth, since it is given by left and right-composition withsmooth maps (see [22] for the noncompact case), and no reparametrization issue ispresent. Consequently, part of the technical low regularity arguments in our equi-variant implicit function theorem are not required here. Due to these symmetries,if K is a Killing vector field of ( M , g λ ), then K ◦ φ is a Jacobi field. Likewise, if K is a Killing field of ( M, g λ ), then φ ∗ ( K ), defined by M ∋ p d φ ( p ) K p ∈ T φ ( p ) M ,is a Jacobi field. Definition 5.6.
The space of Jacobi fields along φ spanned by fields of the type K ◦ φ and φ ∗ ( K ), for Killing fields K and K , is called the space of Killing-Jacobifields along φ . A ( g λ , g λ )-harmonic map φ : M → M is said to be equivariantlynondegenerate if the space of Jacobi fields along φ coincides with the space ofKilling-Jacobi fields along φ . Definition 5.7.
Two harmonic maps φ and φ are called geometrically equiv-alent if they are in the same Iso( M, g λ ) × Iso(
M , g λ )-orbit, i.e., if there exists g ∈ Iso( M, g λ ) and g ∈ Iso(
M , g λ ) such that φ = g ◦ φ ◦ g − . Theorem 5.8.
Let ( g λ , g λ ) be a family of pairs of Riemannian metrics on M and M respectively, parametrized by λ ∈ Λ , for some manifold Λ . Assume that G λ = Iso( M, g λ ) × Iso(
M , g λ ) , λ ∈ Λ , forms a smooth bundle of Lie groups G = { G λ : λ ∈ Λ } . Let φ : M → M be an equivariantly nondegenerate ( g λ , g λ ) -harmonic map. Then, there exists a neighborhood V of λ in Λ , and a smoothfunction φ : V → X satisfying: (a) φ ( λ ) is a ( g λ , g λ ) -harmonic map for all λ ∈ V ; (b) φ ( λ ) = φ .Moreover, given λ ∈ V , any other ( g λ , g λ ) -harmonic map from M to M sufficientlyclose to φ is geometrically equivalent to φ ( λ ) .Proof. All assumptions of the equivariant implicit function theorem are satisfied bythe harmonic maps variational problem, using the following objects: • X ′ = X = C ,α ( M, M ); • E is the mixed vector bundle whose fiber E φ is C ,α ( φ ∗ T M ), the Banachspace of all C ,α -H¨older vector fields along φ , endowed with the topology C ,α on the base and C ,α on the fibers; • the inclusion T X ⊂ E is obvious, and the inner product on the fibers E x is induced by the L -pairing taken with respect to the volume form (ordensity) of a fixed background Riemannian metric g ∗ on M ; • δf ( φ, g ) = − ξ g ∆ g , g ( φ ), where ξ g : M → R is the positive C k functionsatisfying ξ g vol g ∗ = vol g , see (5.4) and (5.6); • Y = T X , κ is the identity map and j is induced by the L -pairing, as above; • identifying the Lie algebra g λ of G λ with the space of pairs ( K, K ) of(complete) Killing vector fields on ( M, g ) and ( M , g ), for each φ ∈ X the map d β λφ (1 λ ) : g λ → T φ X associates to a pair ( K, K ) the vector field K ◦ φ − φ ∗ ( K ) along φ . The image of this map is precisely the space ofKilling-Jacobi fields along φ , and the notion of equivariant nondegeneracyfor harmonic maps coincides with the abstract notion of nondegeneracygiven in our equivariant implicit function theorem; • given a ( g , g )-harmonic map φ : M → M , the vertical projection of thederivative ∂ ( δf )( φ, g ) is identified via (3.2) with ξ g J φ , where J φ is theJacobi operator in (5.7). This is an elliptic second order partial differentialoperator and ξ g J φ : C ,α ( φ ∗ T M ) → C ,α ( φ ∗ T M ) is a Fredholm operatorof index zero, see [28, Thm 1.1, (2)]. (cid:3)
Deformation of Hamiltonian stationary Lagrangian submanifolds.
Let (
M, ω ) be a symplectic manifold with dim M = 2 n and Σ be a compact manifoldwith dim Σ = n . An embedding x : Σ → ( M, ω ) is called
Lagrangian if x ∗ ω = 0.In this case, we say x (Σ) ⊂ M is a Lagrangian submanifold. A smooth family x s : Σ → M , s ∈ ( − ε, ε ), of embeddings is called a Hamiltonian deformation of x = x if its derivative X = dd s x s (cid:12)(cid:12) s =0 is a Hamiltonian vector field along x , i.e., ifthe 1-form σ X := x ∗ ( ω ( X, · )) on Σ is exact.Endow ( M, ω ) with a Riemannian metric g . This metric allows us to computethe volume of an embedding x : Σ → M , by pulling back its volume form vol g .A Lagrangian embedding x : Σ → M is called g -Hamiltonian stationary if it hascritical volume with respect to any Hamiltonian deformations x s : Σ → M , s ∈ ( − ε, ε ), of x . Hamiltonian stationary Lagrangian submanifolds are analogous tominimal submanifolds (i.e., the case H = 0 in the CMC problem described in EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 21
Subsection 5.1), but instead of minimizing volume in all directions, they minimizevolume only in the
Hamiltonian directions. In particular, minimal Lagrangiansubmanifolds are automatically Hamiltonian stationary.We now describe the basis for the variational setup of this problem, followingclosely [3, Sec 4]. Let L (Σ , M ) be the space of C ,α -unparametrized Lagrangianembeddings of Σ in ( M, ω ), i.e., the quotient of the space of Lagrangian embeddingsof Σ into M of class C ,α by the action of the diffeomorphism group of Σ. Thespace L (Σ , M ) can be identified with the set of Lagrangian submanifolds of M ofclass C ,α that are diffeomorphic to Σ. Its structure is analogous to the set ofunparametrized embeddings used in Subsection 5.1 to study CMC hypersurfaces,in that, given a smooth Lagrangian embedding x : Σ → M , there exists a localchart around the unparametrized embedding [ x ] ∈ L (Σ , M ) with values in theBanach space Z (Σ) of closed 1-forms on Σ of class C ,α . In particular, we get anidentification(5.10) T [ x ] L (Σ , M ) = Z (Σ) . An important finite codimension subspace of Z (Σ) is the space B (Σ) of exact1-forms on Σ, of the same regularity. The corresponding distribution, via theidentification (5.10), is an integrable distribution of L (Σ , M ) with codimension b (Σ) = dim H (Σ , R ). This distribution is called Hamiltonian distribution , and itsintegral leaves near [ x ] are parametrized by elements of the first de Rham coho-mology H (Σ , R ) = Z (Σ) /B (Σ). Denote by [ η ] ∈ H (Σ , R ) the cohomology classof a closed 1-form η , and by L (Σ , M ) [ η ] the integral leaf of the Hamiltonian distri-bution corresponding to [ η ]. Also, note that L (Σ , M ) [0] is the space of Hamiltoniandeformations of [ x ].Given a smooth unparametrized Lagrangian embedding [ x ] ∈ L (Σ , M ), let U bean open neighborhood of [ x ], that is identified with a neighborhood of 0 ∈ Z (Σ).Shrinking U if necessary, suppose that the natural splitting Z (Σ) = B (Σ) ⊕ H (Σ , R ) induces a product structure U = U B × U H , where U B is a neighborhoodof 0 ∈ B (Σ) and U H is a neighborhood of [0] ∈ H (Σ , R ). Under this identification,each [ x ] ∈ U corresponds to a unique pair ( ζ, [ η ]) ∈ U B × U H , and [ x ] correspondsto (0 , [0]).Set X = U B and Λ = U H . Let g t be a family of metrics on M of class C k , k ≥
4, parametrized by t ∈ Λ , where Λ is a manifold. Consider the space ofparameters Λ = Λ × Λ , which is a finite-dimensional manifold formed by pairs λ = ([ η ] , t ). Let(5.11) f : X × Λ → R , f ( ζ, [ η ] , t ) = Z Σ x ∗ (vol g t ) , where x is a Lagrangian embedding that represents the class [ x ] = ( ζ, [ η ]) ∈ U = U B × U H and vol g t is the volume form of ( M, g t ). It is easy to verify that the aboveis well-defined and, for a given λ = ([ η ] , t ) ∈ Λ, critical points of f λ ( ζ ) = f ( ζ, λ )are precisely the g t -Hamiltonian stationary Lagrangian submanifolds that belongto L (Σ , M ) [ η ] . This is just a somewhat intricate way of (locally) encoding the factthat the g t -Hamiltonian stationary Lagrangian submanifolds that belong to the leaf L (Σ , M ) [ η ] of the Hamiltonian distribution are the critical points of the g t -volumefunctional restricted to L (Σ , M ) [ η ] . More precisely, for a Hamiltonian variation X , with σ X = d h , of the embedding [ x ] = ( ζ, [ η ]),(5.12) ∂ f ( ζ, [ η ] , t ) h = Z Σ g t ( ~H, X ) = Z Σ g t ( σ H , d h ) = Z Σ (d ∗ σ H ) h, where ~H is the mean curvature vector of x and d ∗ is the codifferential. Here, weidentify the tangent space to U B at ζ as B (Σ) = C ,α (Σ) / R , i.e., an exact 1-formd h of class C ,α is identified with the function h modulo constants.Under the extra assumption that ( M, ω, g t ) is a K¨ahler manifold, with com-plex structure J t , the Jacobi operator at a g t -Hamiltonian stationary Lagrangianembedding x = ( ζ, [ η ]) is given by the following formula (see [13, (9), p. 1072]):(5.13) J ζ ( h ) = ∆ h + d ∗ σ Ric ⊥ ( J t ∇ h ) − ∗ σ B ( J t ~H, ∇ h ) − J t ~H ( J t ~H ( h )) , where B is the second fundamental form of x (Σ) ⊂ M and Ric ⊥ ( X ) for a normalvector X to Σ is defined by g t (Ric ⊥ ( X ) , Y ) = Ric( X, Y ) for all Y normal to Σ.Again, the actual linear operator that represents the second variation of the abovefunctional is defined in the tangent space to U B , which is C ,α (Σ) / R . In otherwords, (5.13) induces an operator [ J ζ ] : C ,α (Σ) / R → C ,α (Σ) / R , since it acts ond h and we are representing it by its action on h . Exact 1-forms η on Σ that are inthe kernel of the Jacobi operator at x are called Jacobi fields along x .Assume that G t = Iso( M, g t ) acts on ( M, ω ) in a Hamiltonian fashion, see [3,Sec 2]. As in the above cases, the functional f is invariant under this action andthere are special Jacobi fields that come from the symmetry group. Namely, if K is a Killing field on ( M, g t ), then the 1-form σ K = x ∗ ( ω ( K, · )) is exact, because theaction is Hamiltonian. Moreover, since the action is by symplectomorphisms and g t -isometries, σ K is a Jacobi field. Definition 5.9.
Jacobi fields of the form σ K as above, where K is a Killing field, arecalled Killing-Jacobi fields . The g t -Hamiltonian stationary Lagrangian embedding x is called equivariantly nondegenerate if the space of Jacobi fields along x coincideswith the space of Killing-Jacobi fields along x . Definition 5.10.
Two Hamiltonian stationary Lagrangian embeddings x i : Σ → M are said to be congruent if there exists a diffeomorphism φ of Σ and an isometry F ∈ G t , such that x = F ◦ x ◦ φ . Theorem 5.11.
Let ( M, ω ) be a symplectic manifold and g t be a family of Rie-mannian metrics on M , parametrized by t ∈ Λ , for some manifold Λ . Assumethat G t := Iso( M, g t ) forms a smooth bundle of Lie groups G = { G t : t ∈ Λ } ,and that the G t -action on ( M, ω ) is Hamiltonian. Suppose g t is a K¨ahler met-ric and x : Σ → M is an equivariantly nondegenerate g t -Hamiltonian stationaryLagrangian embedding. Then, there exists a neighborhood V of [0] ∈ H (Σ , R ) , aneighborhood V of t ∈ Λ , a neighborhood W of [ x ] in L (Σ , M ) and a smoothmap x : V → W , from the neighborhood V = V × V of ([0] , t ) to W , satisfying: (a) x ([ η ] , t ) is a g t -Hamiltonian stationary Lagrangian embedding which is in L (Σ , M ) [ η ] , for all ([ η ] , t ) ∈ V ; (b) x ([0] , t ) = x .Moreover, given ([ η ] , t ) ∈ V , any other g t -Hamiltonian stationary Lagrangian em-bedding in L (Σ , M ) [ η ] sufficiently close to x is congruent to x ( λ ) . i.e., the formal adjoint of the exterior derivative operator d, with respect to g t . EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 23
Proof.
Similarly to Theorem 5.2, this is an application of the non-regular versionof the equivariant implicit function theorem discussed in Section 4. As above,consider a local chart U = U B × U H ⊂ B (Σ) × H (Σ , R ) of [ x ] ∈ L (Σ , M ), where[ x ] corresponds to (0 , [0]), and set the manifold X to be the neighborhood U B of theorigin in the space of exact 1-forms on Σ of class C ,α . The space of parameters isΛ = U H × Λ , and the functional f is given by (5.11). Critical points of f λ : X → R , λ = ([ η ] , t ), are the (classes of) g t -Hamiltonian stationary Lagrangian embeddingsthat belong to L (Σ , M ) [ η ] .Analogously to the CMC setup, the map β tζ : G t → X is smooth when [ x ] =( ζ, [ η ]) is in the dense subset of (classes of) smooth embeddings. The image ofd β tζ (1 t ) is the space of Killing-Jacobi fields, so that the equivariant nondegeneracyassumption on [ x ] is precisely the equivariant nondegeneracy assumption of ourimplicit function theorem.The remaining objects used to apply the abstract implicit function theorem are: • E = C ,α (Σ) / R is the Banach space of C ,α functions on Σ modulo con-stants. The tangent space to T ζ U B is canonically identified with the spaceof exact 1-forms on Σ of class C ,α , hence the inclusion T X ⊂ E is by firstidentifying such an exact form d h with the C ,α function h , and then usingthe inclusion C ,α ֒ → C ,α ; • The pairing h· , ·i on E is the natural L -pairing of functions, h h , h i = R Σ h h vol g ∗ , with respect to the volume form (or density) of a fixed back-ground Riemannian metric g ∗ ; • Y = C ,α (Σ) / R , the bundle morphism κ : T X → Y is the obvious inclusionand j : E → Y ∗ is induced by the L -pairing above; • Given λ = ([ η ] , t ), identifying the Lie algebra g t of G t with the spaceof (complete) Kiling vector fields on ( M, g t ), when ζ ∈ U B is such that[ x ] = ( ζ, [ η ]) is smooth, then d β tζ (1 t ) : g t → T ζ U B associates to a Killingfield K the exact 1-form σ K ; • δf λ ( ζ ) = ξ g t d ∗ σ H , where σ H = x ∗ ( ω ( H, · )) and x is an embedding in theclass of [ x ] = ( ζ, [ η ]), if λ = ([ η ] , t ) and ξ g t is the positive function satisfying ξ g t vol g ∗ = vol g t , see (5.12); • the vertical derivative of δf λ at ζ is identified via (3.2) with the Jacobioperator [ J ζ ] induced by (5.13). By this formula, J ζ is a linear elliptic op-erator of fourth-order, hence Fredholm of index 0 from C ,α (Σ) to C ,α (Σ).The induced operator modulo constants, [ J ζ ] : C ,α (Σ) / R → C ,α (Σ) / R ,remains Fredholm of index 0, see [3, Sec 4.4]. (cid:3) Appendix A. Symmetric Fredholm operators
Consider the following setup: • X and Y are Banach spaces; • H is a Hilbert space, with inner product h· , ·i ; • there exist continuous injections X ֒ → Y ֒ → H with dense images; • T : X → Y is a (bounded) Fredholm operator; • T is symmetric, i.e., h T x , x i = h x , T x i for all x , x ∈ X .It is natural to conjecture that, with the above hypotheses, the Fredholm index of T is zero. However, this is false in general, as shown by the following counter-example. Example.
Set H = L ([0 , ⊕ R , and Y = H . Let u : ]0 , → R be the function u ( x ) = x , and let D ⊂ L ([0 , D = { f ∈ L ([0 , f u ∈ L ([0 , } , endowed with the inner product h f, g i ∗ = h f, g i L + h f u, gu i L . In other words, this is the inner product that makes D isometric to thegraph of the linear operator D ∋ f f u ∈ L ([0 , D is a Hilbert space when endowed with h· , ·i ∗ .Let α : D → R be the bounded linear functional α ( f ) = R f u . Clearly, α iscontinuous in the topology induced by h· , ·i ∗ , but discontinuous in the topologyof H . Let X ⊂ D ⊕ R be the graph of α . This is a closed subspace of D ⊕ R ,when D is endowed with the inner product h· , ·i ∗ , and therefore it is a Hilbertspace. Moreover, X is dense in D ⊕ R (hence in Y = H ) when D is endowed withthe topology induced by H , because it is the kernel of the unbounded functional D ⊕ R ∋ ( f, t ) t − α ( f ) ∈ R .Let S : D ⊕ R → H the linear map defined by S ( f, c ) = ( f u, S issymmetric relatively to the inner product of H , and continuous when D is endowedwith the topology induced by h· , ·i ∗ . The kernel of S is { } ⊕ R , and the imageof S is L (cid:0) [0 , (cid:1) ⊕ { } . Let T : X → H be the restriction of S to X . Also T is symmetric relatively to the inner product of H and continuous in the topologyinduced by h· , ·i ∗ . The kernel of T is X ∩ ( { } ⊕ R ) = { } , and the image of T isagain L (cid:0) [0 , (cid:1) ⊕ { } , which is a closed subspace of H having codimension 1. Thus, T is a Fredholm operator of index − X ֒ → Y ֒ → H ,which provide continuous inclusions of the dual spaces: H ∗ ֒ → Y ∗ ֒ → X ∗ . Identi-fying H with H ∗ , we have the chain of inclusions X ֒ → Y ֒ → H = H ∗ ֒ → Y ∗ ֒ → X ∗ . Due to these inclusions and the symmetry of T , the adjoint operator T ∗ : Y ∗ → X ∗ is an extension of T . Proposition.
Let X , Y , H , and T : X → Y be as above. In addition, assume thatthe following ellipticity hypothesis is satisfied: (A.1) For all x ∈ Y ∗ , T ∗ ( x ) ∈ Y implies x ∈ X, i.e., ( T ∗ ) − ( Y ) ⊂ X . Then, the Fredholm index of T is zero.Proof. It suffices to show that Y is the direct sum of Im( T ) and ker( T ), whereIm( T ) is the image of T . First, the symmetry of T implies that Im( T ) and ker( T )are orthogonal, hence Im( T ) ∩ ker( T ) = { } . Given any y ∈ Y , let us show that y ∈ ker( T ) + Im( T ). Let V denote the closure of Im( T ) in H , and let z be theorthogonal projection of y onto V . Then, y − z ∈ H , it is orthogonal to Im( T ),thus T ∗ ( y − z ) = 0. By hypothesis (A.1), y − z ∈ X and T ( y − z ) = 0, therefore y − z ∈ ker( T ). It remains to show that z ∈ Im( T ). Since y ∈ Y and y − z ∈ X ,then z ∈ Y . Moreover, z ∈ V , therefore z is orthogonal to ker( T ), i.e., the linearfunctional h z, ·i annihilates ker( T ). Since Im( T ) is closed, the annihilator of ker( T )is precisely the image of T ∗ , and therefore there exists w ∈ Y ∗ such that T ∗ ( w ) = z .But z ∈ Y , and by (A.1), w ∈ X , hence z ∈ Im( T ). (cid:3) We conclude with a remark on the interpretation of the “ellipticity” assump-tion (A.1). In concrete examples, H is an L -space, T is a formally self-adjoint EFORMING VARIATIONAL PROBLEMS WITH VARYING SYMMETRIES 25 differential operator between Banach spaces X and Y , of functions having a cer-tain regularity, and T ∗ will be the extension of T to some space of distributions.Given y ∈ Y , weak solutions of the equality T ( x ) = y are vectors x ∈ Y ∗ satisfying T ∗ ( x ) = y . Assumption (A.1) means that, when y ∈ Y , weak solutions x ∈ Y ∗ of the equality T ( x ) = y also belong to X ; hence are are in fact true solutions of the equation. This hypoellipticity property is satisfied, e.g., by linear ellipticdifferential operators with the appropriate choices of functions spaces X and Y . References [1]
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