Sobolev metrics on spaces of manifold valued curves
aa r X i v : . [ m a t h . DG ] J u l SOBOLEV METRICS ON SPACES OF MANIFOLDVALUED CURVES
MARTIN BAUER, CY MAOR, AND PETER W. MICHOR
Abstract.
We study completeness properties of reparametrization in-variant Sobolev metrics of order n ≥ Contents
1. Introduction and main results 12. Spaces of manifold valued functions and immersions 63. Reparametrization invariant Sobolev metrics on spaces of curves 104. Estimates 155. Metric and geodesic completeness 166. Incompleteness of constant coefficient metrics on open curves 27Appendix A. The geodesic equation 32Appendix B. Holonomy estimates: proof of Lemma 4.2 37Appendix C. Proof of Lemma 5.12 43References 451.
Introduction and main results
Background.
In recent years Riemannian geometry on the space ofcurves has been an area of active research. The motivation for these in-vestigations can be found in the area of shape analysis, where the space ofgeometric curves plays an important role: closed planar curves are used toencode the outlines (shapes) of planar objects, and elastic (reparametriza-tion invariant) Riemannian metrics have been successfully used to comparethese objects in a variety of different applications [32, 33, 38, 39]. Morerecently, curves with values in a manifold have emerged as a topic of in-terest in shape analysis as well. Examples include the study of trajectories
Mathematics Subject Classification. on the earth [34, 35], of computer animations [18], or of brain connectivitydata [19]. Here the brain connectivity of a patient over time is representedas a path in the space of positive, symmetric matrices. Motivated by theseapplications several of the numerical algorithms, as originally developed forplanar curves, have been generalized to this more complicated situation.In this article we are interested in the mathematical properties of theseRiemannian metrics and in particular in questions related to completenessof the corresponding geodesic equations. These investigations build up onclassical questions related to diffeomorphism groups, as reparametrizationinvariant metrics on spaces of immersions can be viewed as generalizationsof right-invariant metrics on diffeomorphism groups. These have been in thefocus of intense research due to their relations to many prominent PDEs viaArnold’s approach to hydrodynamics [1, 2, 36]. Local well-posedness in thissetup was established for a wide variety of invariant metrics, typically usingan Ebin-Marsden type analysis [20, 29, 23, 28, 7]. The focus of this articleis geodesic and metric completeness, which is well understood for strongenough metrics in the case of diffeomorphism groups [39, 30, 29, 16, 5], but ismostly open for spaces of immersions. For closed, regular curves with valuesin Euclidean space, a series of completeness results both on the space ofparametrized and unparametrized curves has been obtained, beginning withBruveris, Michor and Mumford [14], see also [12, 15, 8]. The goal of thisarticle is to generalize these results to the case of open and closed, regularcurves with values in a Riemannian manifold. While the manifold structureof the target space is of little relevance for the local results mentioned before,it significantely complicates the analysis for the global results studied inthe present article. We will comment on the differences to the Euclideansituation in Section 1.4 below; first we describe the main contributions ofthe present article.1.2.
Main Result.
To formulate our main result we first introduce themanifold of regular curves and the class of Riemannian metrics, that wewill consider in this article. For n ≥
2, we consider the space of Sobolevimmersions from a one-dimensional parameter space D with values in acomplete Riemannian manifold with bounded geometry ( N , g ):(1.1) I n ( D, N ) = (cid:8) c ∈ H n ( D, N ) : c ′ ( θ ) = 0 , ∀ θ ∈ D (cid:9) . Here D = [0 , π ] for open curves and D = S for closed curves. The Sobolevspace H n ( D, N ) is defined in more detail in Section 2; note that H n ( D, N ) ⊂ C ( D, N ), hence the condition c ′ ( θ ) = 0 is well defined. On this space wecan consider reparametrization invariant (elastic) Sobolev metrics of order n . The class we focus on in this paper is given by G c ( h, k ) = n X i =0 a i ( ℓ c ) Z D g ( ∇ i∂ s h, ∇ i∂ s k ) d s, (1.2)where a i ∈ C ∞ ((0 , ∞ ) , [0 , ∞ )), ∇ is the covariant derivative in N , s = | c ′ | is the norm of c ′ with respect to the Riemannian metric g , i.e., the arc-lenth function along c . Furthermore, d s = | c ′ | d θ is the arc length one form, ∂ s = | c ′ | ∂ θ is the arc length vector field along the curve, and ℓ c = R D d s is OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 3 the length of the curve. The two most important sub-families of these typeare (1) the constant coefficient
Sobolev metrics, where a i ( ℓ c ) = C i ≥ ℓ c ;(2) the family of scale invariant Sobolev metrics where a i ( ℓ c ) = C i ℓ n − c with C i ≥ N is the Euclidean space, composition with rescaling x αx of the target manifold is an isometry of ( I n ( D, N ) , G ), for each α > C and C n are strictly positive, to avoid degen-eracy. The main focus of the present article lies on completeness propertiesof these Riemannian metrics. In a slightly simplified version our main resultscan be summarized as follows: Theorem (Main Theorem) . Let D = [0 , π ] or D = S , and let G be thescale invariant Sobolev metric (1.2) of order n . The following completenessproperties hold: (1) ( I n ( D, N ) , dist G ) is a complete metric space. (2) ( I n ( D, N ) , G ) is geodesically complete (3) Any two immersions in the same connected component of ( I n ( D, N ) , G ) can be joined by a minimizing geodesic.For D = S the results continue to hold for the family of constant coefficientSobolev metrics. Previously this result was only known for closed curves in Euclidean space(see [12] for constant coefficients and [16] for a wider class that includesscale invariant ones), and thus the results of the present article generalizethese previous works in two important directions (open curves and curveswith values in a manifold). In fact, we will prove these statements for awider class of metrics, see Theorems 5.1–5.3. Note that in this infinitedimensional situation the theorem of Hopf-Rinow is not valid [3] and thusitem (3) does not follow directly from the metric completeness, but has tobe proven separately.1.3.
Further contributions of the article.
In the following we describeseveral further key contributions of the current article: • Completeness in the smooth setting:
In the main theoremabove, we have formulated the results only in the Sobolev cate-gory. Using an Ebin-Marsden type no-loss-no-gain result [20], weshow that geodesic completeness (i.e., global existence of geodesics)extends to the space of smooth, closed curves (Corollary 5.13). Foropen curves, we only obtain regularity in the interior of the curve,as explained in Section 5.6. • Metric incompleteness of constant coefficient metrics onopen curves:
In [4] it was observed that the space of open curves,with respect to constant coefficient Sobolev metrics, is metrically in-complete; indeed, in the same paper the authors constructed a pathof immersed curves, whose lengths tend to zero after finite time. InSection 6 we elaborate on this example, and show that vanishing of
MARTIN BAUER, CY MAOR, AND PETER W. MICHOR the entire curve is the only way a path (or a sequence) of immersedcurves can leave the space of immersions I n ([0 , π ] , N ) in finite time(Theorem 6.3). That is, a path cannot leave the space by some ”lo-cal” failure, say, by losing the immersion property at a point (sucha failure of completeness can occur in lower-order metrics, e.g., inshockwaves in the inviscid Burgers equation). We give some evi-dence that the completion of the space of open curves in this caseis a one-point completion, where the additional point represents allthe Cauchy sequences converging to vanishing length curves. • Existence of minimizing geodesics for constant coefficientmetrics on open curves:
We show that if the distance betweentwo open curves is lower than some explicit threshold dependingonly on their lengths, then they can be connected by a minimiz-ing geodesic (Theorem 6.7). We note, however, that this thresholdis not necessarily sharp; in fact, in view of the rather rigid way inwhich curves can leave the space, we cannot rule out that a mini-mizing geodesic exists between any two immersions. We also do notknow whether geodesics (unlike general paths of finite length) maycease to exist after finite time, that is, we do not know if the spaceis geodesically incomplete (only that it is metrically incomplete).These questions will be considered in future works. • Local well-posedness:
Our completeness results are only valid formetrics of order n ≥
2, and it can be shown that metrics of lowerorder can never have these properties. Nevertheless, using an Ebin-Marsden type approach, we show local well-posedness for all smoothmetrics of the type (1.2) of order n ≥
1, see Theorem 3.8. Thisresult was previously known for closed curves and the case of opencurves requires some additional considerations for dealing with theboundary terms that appear in the geodesic equation. • Completeness of the intrinsic metric on H n ( D, N ) : It is wellknown that H n ( D, N ), for n > dim D/
2, is a Hilbert manifold, andthat its topology coincides with the one induced via the inclusion H n ( D, N ) ⊂ H n ( D, R m ) that is defined by a closed isometric em-bedding ι : N → R m . This inclusion also induces a complete metricspace structure on H n ( D, N ). As part of the proof of the maintheorem, we show the natural Riemannian metric on H n ( D, N ),(1.3) H c ( h, k ) := Z D g c ( h, k ) + g c ( ∇ n∂ θ h, ∇ n∂ θ k ) d θ, is also metrically complete (Proposition 2.2), thus defining a com-plete metric space structure that is intrinsic (independent of an iso-metric embedding). We study these different definitions and equiv-alence of H n ( D, N ), in Section 2.1.4. Main ideas in the proof and structure of the article.
The tech-niques used in the proof of our main theorem, Theorems 5.1–5.3, expandupon the ones used to study completeness of Euclidean curves [12]. Themain diffuclties arise from taking into account the more complicated struc-ture of the space I n ( D, N ) and the effects of the curvature of N on various OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 5 estimates (in particular, on the behavior of some Sobolev interpolation in-equalities). To give the reader a first glimpse, we will outline the strategyand main steps below.
Local well-posedness . As a basis to the rest of the analysis, we first studythe metric G (as in (1.2)) in Section 3, and prove that it is a smooth,strong metric on I n ( D, N ). In this section we also give some details on theassociated geodesic equation and formulate the local well-posedness result(as this theorem is not the focus of the present article, we postpone its proofto the appendix A.2).
Metric and geodesic completeness . The space of Sobolev immersions I n ( D, N ) is an open subset of H n ( D, N ), which is metrically complete withrespect to the metric H , defined in (1.3); this is established in Section 2.Note that for N = R d this is trivial, as H n ( D, N ) is a Hilbert space in thiscase.Since ( H n ( D, N ) , dist H ) is a complete metric space, showing metric com-pleteness of ( I n ( D, N ) , dist G ) can be reduced to showing that G and H areequivalent metrics, uniformly on every dist G -ball in I n ( D, N ), and that thespeed | c ′ | of an immersion c ∈ I n ( D, N ) is bounded away from zero on everydist G -ball. This reduction is done in detail in Section 5.1.In order to obtain the uniform equivalence of G and H on metric balls, oneneeds to obtain bounds on the length ℓ c of the curve, and on certain normsof the velocity c ′ , uniformly for all immersions c in a metric ball. This isdone in Section 5.2, and the proof of metric completeness is then concludedin Sections 5.3–5.4. As metric completeness implies geodesic completeness ofstrong Riemannian metrics also in infinite dimensions, see [24, VIII, Propo-sition 6.5], this also concludes the proof of geodesic completeness.The main technical tool for establishing the bounds on ℓ c and c ′ areSobolev interpolation inequalities on the tangent space T c I n ( D, N ), withexplicit dependence of the inequalities constants on the length of the basecurve c . In the case of closed curves, there is non-trivial holonomy along thecurves, hence we need to control the holonomy along a curve in terms of itslength, and apply these estimates to the interpolation inequalities (this isone of the main technical differences from the Euclidean case). These aredone in Section 4, though some of the geometric estimates are postponed toAppendix B. Existence of minimal geodesics . To prove existence of minimal geodesicsbetween two immersions c and c , we consider the energy of paths c t :[0 , → I n ( D, N ) between c and c (defined by the metric G ), and usethe direct methods of the calculus of variations to prove that a minimizingsequence of paths converges, in an appropriate sense, to an energy minimizer(which is, by definition, a geodesic). This is done in Section 5.5. Sincethis approach relies heavily on weak convergence of paths, and the weaktopology is not readily available on the Hilbert manifold I n ( D, N ), we first A Riemannian metric G on a manifold M is a section of non-degenerate bilinear formson the tangent bundle. A strong Riemmanian metric also satisfies that for each x ∈ M ,the topology induced by G x on T x M coincides with the original topology (induced by themanifold structure) on T x M . If dim M < ∞ , every metric is a strong one, but in infinitedimensions this is not the case. MARTIN BAUER, CY MAOR, AND PETER W. MICHOR embed it into the Hilbert space H n ( D, R m ) via a closed isometric embedding ι : N → R m . The analysis then combines the same type of bounds that areused to prove metric completeness, with bounds that relate the metric on H n ( D, R m ) to the metric G on I n ( D, N ) (similar bounds are also used inproving the completeness of H n ( D, N ) with respect to the dist H metric inSection 2). Acknowledgements.
We would like to thank to Martins Bruveris, FXVialard and Amitai Yuval for various discussions during the work on thispaper.2.
Spaces of manifold valued functions and immersions
Let ( N , g ) be a (possibly non-compact) complete Riemannian manifoldwith bounded geometry, where the induced norm of the Riemannian metricwill be denoted by | · | = p g ( · , · ). We will denote its covariant derivative by ∇ , or, where ambiguity might arise, by ∇ N . With a slight abuse of notation,we will also use it as the covariant derivative on pullbacks of T N .We consider the space of (closed or open) regular curves with values in N , which we denote by(2.1) Imm( D, N ) = (cid:8) c ∈ C ∞ ( D, N ) : c ′ ( θ ) = 0 , ∀ θ ∈ D (cid:9) . Here D = S for closed curves and D = [0 , π ] for open curves. This spaceis an infinite dimensional manifold, whose tangent space at a curve c is thespace of vector fields along c :(2.2) T c Imm( D, N ) = { h ∈ C ∞ ( D, T N ) : π ( h ) = c } , where π denotes the foot point projection from T N to N .To obtain the desired completeness and well-posedness results we needto consider a larger space metric of Sobolev immersions I n ( D, N ) ⊃ Imm( D, N ), for n ≥
2, which we define below.
Definition 2.1.
Let N be a Riemannian manifold as above, and fix a proper,smooth, isometric embedding ι : N → R m , for large enough m ∈ N . For n ≥
2, we define the Sobolev space H n ( D, N ) and the space of Sobolevimmersions I n ( D, N ) as follows:(1) H n ( D, N ) consists of all maps c : D → N such that ι ◦ c ∈ H n ( D ; R m ).(2) I n ( D, N ) consists of all c ∈ H n ( D, N ) such that c ′ ( θ ) = 0 , ∀ θ ∈ D .With this (extrinsic) definition of H n ( D, N ), it inherits the metric struc-ture of H n ( D ; R m ), which we denote by dist ext ; since convergence in thespace H n ( D ; R m ) implies uniform convergence, we have that H n ( D, N ) is aclosed subset of H n ( D ; R m ), hence a complete metric space with respect todist ext . We are interested in characterizing H n ( D, N ) as an infinite dimen-sional Riemannian manifold. The main goal of this section is to prove thefollowing: Proposition 2.2.
The space H n ( D, N ) , ≤ n ∈ N is a Hilbert manifoldwhose tangent space at c is H n ( D ; c ∗ T N ) . Moreover, it is a complete met-ric space with respect to the distance function dist H induced by the smooth OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 7
Riemannian metric (1.3) : H c ( h, k ) := Z D g c ( h, k ) + g c ( ∇ n∂ θ h, ∇ n∂ θ k ) d θ. Finally, the space of Sobolev immersions I n ( D, N ) is an open subset of H n ( D, N ) and in particular is a Hilbert manifold with the same tangentspace. Henceforth, we will always associate H n ( D, N ) with the metric dist H (rather than dist ext ). Note that, in general, dist H and dist ext are not equiv-alent metrics. Note also that for h ∈ T c I n ( D, N ) there are two natural L metrics: in one we integrate with respect to d θ , and in the other withrespect to arc length d s = | c ′ | d θ ; we denote the first one by L ( d θ ) and thesecond by L ( d s ).Proposition 2.2 holds for much more general manifold domain D : namely,the Hilbert manifold structure exists whenever 2 n > dim D , and the open-ness of I n ( D, N ) in H n ( D, N ) holds whenever 2( n − > dim D . Theseare known results and we describe their proofs below for completeness. Tothe best of our knowledge, the completeness of ( H n ( D, N ) , dist H ) has notbeen considered before; we expect it to hold, again, whenever 2 n > dim D ,virtually with the same proof as the one below (using H¨older inequalitiesinstead of uniform bounds). Proof.
Part I: Smooth structure and topology
An alternative charac-terization of H n ( D, N ) is H n ( D, N ) = n c ∈ C ( D, N ) : c = exp s ( V )for some s ∈ C ∞ ( D, N ) , V ∈ H n ( D ; s ∗ T N ) o , where exp is the exponential map with respect to the Riemannian metric g on N (see, e.g., [37, Lemma B.5]). This characterization induces a smoothstructure on H n ( D, N ), where the charts, modeled on H n ( D ; s ∗ T N ), aregiven by exp s for s ∈ C ∞ ( D, N ). The tangent space at c is H n ( D ; c ∗ T N ).See [27, 5.3–5.8] for details. This smooth structure is described in detail in[22, Section 3] (it is denoted there by A sg ). In [22, Proposition 3.7] it is shownthat this smooth structure coincides with the one induced by consideringlocal charts on D and N (which provides yet another characterization to H n ( D, N )).Next, note that the topology induced by this smooth structure is equiva-lent to the topology induced on H n ( D, N ) by dist ext [37, Lemma B.7]. Theinner product H c describes the Hilbert space topology on the tangent space T c H n ( D, N ) = H n ( D ; c ∗ T N ). Since these are also the modeling spaces forthe natural chart construction, H is a strong Riemannian metric. Thus thedistance function dist H induced by H induces the topology of H n ( D, N ). Part II: Openness of I n ( D, N ) in H n ( D, N ) Taking again the extrinsicpoint of view I n ( D, N ) is the intersection of H n ( D, N ) with all the maps c ∈ H n ( D ; R m ) such that c ′ = 0. By the Sobolev embedding k c ′ k L ∞ ( D ; R m ) ≤ C k c k H n ( D ; R m ) , which holds since n ≥
2, it is immediate that c ′ = 0 is anopen condition in H n ( D ; R m ), and hence I n ( D, N ) is open in H n ( D, N ). MARTIN BAUER, CY MAOR, AND PETER W. MICHOR
Part III: Completeness of ( H n ( D, N ) , dist H ) Let c j ∈ H n ( D, N ) be aCauchy sequence with respect to dist H . We aim to show that c j is also aCauchy sequence with respect to dist ext . Then, since ( H n ( D, N ) , dist ext )is complete, we will obtain that the sequence converges to some c ∞ ∈ H n ( D, N ); since the topologies induced by dist ext and dist H coincide, wewill obtain that ( H n ( D, N ) , dist H ) is complete as well.Since ( c j ) j ∈ N is a dist H -Cauchy sequence, it lies inside some dist H -ball B of radius r > c ∈ H n ( D, N ). By taking a slightly larger r we can also assume that for every j ≤ k ∈ N there exists a path c jk : [0 , → H n ( D, N ) connecting c j and c k (that is, c jk (0) = c j and c jk (1) = c k ), suchthat c jk ( t ) ∈ B for every t ∈ [0 ,
1] and L H ( c jk ) < dist H ( c j , c k ) + j , where L H is the length of c jk with respect to the metric H .We now show that all the curves in B lie inside a compact subset of N ;moreover, we show that for some C >
0, all curves c ∈ B satisfy k∇ k∂ θ c ′ k L ( dθ ) < C, k = 0 , . . . , n − . It then follows by Lemma 2.3 that there exists a constant β >
0, such thatfor every c ∈ B and every h ∈ H n ( D ; c ∗ T N ), k ι ∗ h k H n ( D ; R m ) ≤ β k h k H c , where ι ∗ h ∈ H n ( D ; R m ) is the image of h under the embedding, and where k · k H n ( D ; R m ) is the standard norm in H n ( D ; R m ) (see Lemma 2.3 below).Therefore, for every j ≤ k ∈ N ,dist ext ( c j , c k ) ≤ L ext ( c jk ) ≤ βL H ( c jk ) < β (cid:18) dist H ( c j , c k ) + 1 j (cid:19) , where L ext is the length with respect to the external structure. Thus ( c j ) j ∈ N is a dist ext -Cauchy sequence and the proof is complete.It remains to verify the assumptions of Lemma 2.3. Let now ¯ c ∈ B = B ( c , r ). By definition, there exists a path c : [0 , → H n ( D, N ), with c (0) = c and c (1) = ¯ c such that L H ( c ) < r . Now, for every θ ∈ D , we havedist N ( c ( θ ) , ¯ c ( θ )) ≤ Z | ∂ t c ( t, θ ) | d t ≤ Z k ∂ t c k L ∞ ≤ C Z k ∂ t c t k H c = CL H ( c ) < Cr, where we used the Sobolev embedding on vector bundles as in Lemma 4.1.It follows that the images of all the curves in B lie in a compact subset of N (namely a neighborhood of radius Cr around the image of c ). OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 9
Now, let k = 0 , . . . , n −
1, then k∇ k∂ θ ¯ c ′ k L ( d θ ) − k∇ k∂ θ c ′ k L ( d θ ) = Z ∂ t (cid:18)Z D |∇ k∂ θ c ′ | d θ (cid:19) / d t = Z R D g ( ∇ k∂ θ c ′ , ∇ k∂ θ ∂ t c ′ ) d θ (cid:16)R D |∇ k∂ θ c ′ | d θ (cid:17) / d t ≤ Z (cid:18)Z D |∇ k∂ θ ∂ t c ′ | d θ (cid:19) / ≤ C Z k ∂ t c k H c d t = L H ( c ) < Cr, where we used Lemma 4.1 again. The uniform bound on k∇ k∂ θ ¯ c ′ k L ( d θ ) immediately follows, and thus the assumptions of Lemma 2.3 are fulfilled,uniformly on B . (cid:3) We now prove the technical lemma that was used above, that showsthe local equivalence of the H c norm and the restriction of the standard H n ( D ; R m ) norm. This lemma will be used again later, when we proveexistence of minimizing geodesics between immersions in Section 5.5. Lemma 2.3.
Let ι : N → R m be an isometric embedding, and let n ≥ .Let K ⊂ N be a compact set, and let c ∈ H n ( D, N ) be a curve whose imagelies in K . Let C > be such that k∇ k∂ θ c ′ k L ( d θ ) < C, k = 0 , . . . , n − . For h ∈ H n ( D ; c ∗ T N ) , denote by ι ∗ h ∈ H n ( D ; R m ) the image of h underthe embedding. The extrinsic norm of h is then defined by k h k H n ( ι ) := k ι ∗ h k H n ( D ; R m ) = Z π | ι ∗ h | + | ∂ nθ ι ∗ h | d θ, where | · | is the norm in R m , and ∂ θ = ∇ R N c ′ is the standard derivative on R m . Then, there exists a constant β > , depending only on ι , K and C such that for every h ∈ H n ( D ; c ∗ T N ) , β − k h k H n ( ι ) ≤ k h k H c ≤ β k h k H n ( ι ) . Proof.
First, note that by standard Sobolev estimates (see Lemma 4.1 foran exact statement), we have that our bounds on k∇ k∂ θ c ′ k L ( d θ ) imply that k∇ k∂ θ c ′ k ∞ < C, k = 0 , . . . , n − , by possibly enlarging the constant C .Next, note that k h k L ( ι ) = k h k L ( d θ ) , since ι is an isometric embedding | ι ∗ h | = | h | pointwise for every θ (here, the R m -norm appears on the lefthandside, the T N -norm on the righthand side).Denote by II the second fundamental form of N in R m , that is, for v, w ∈ T x N , we have II( v, w ) = ∇ R m v w − ∇ N v w. Such a constant exists by Lemma 4.1, since c ′ ∈ H n − . In a coordinate patch on a tubular neighborhood of N , with coordinates( x i ) mi =1 such ( x a ) da =1 , where d = dim N are coordinate on N and ∂ x α ⊥ ∂ x a for a = 1 , . . . , d , α = d + 1 , . . . , m , we haveII( v, w ) = Γ αab ( x ) v a w b ∂ α , where Γ kij are the Christoffel symbols of ∇ R m in these coordinates. Since ∇ N v w ⊥ II( v, w ), we have | ∂ θ ι ∗ h | = |∇ N ∂ θ h | + | II( c ′ , h ) | ≤ |∇ N ∂ θ h | + C | II | | h | ≤ |∇ N ∂ θ h | + C ′ | h | , (2.3)where C ′ = C sup x ∈ K | II | . Integrating, we obtain k∇ N ∂ θ h k L ( d θ ) ≤ k ∂ θ ι ∗ h k L ( d θ ) ≤ k∇ N ∂ θ h k L ( d θ ) + C ′ k h k L ( d θ ) . k h k H ( d θ ) . For the second order terms we calculate ∂ θ ι ∗ h = ∂ θ ∇ N ∂ θ h + ∂ θ (II( c ′ , h )) = ( ∇ N ∂ θ ) h + II( c ′ , ∇ N ∂ θ h ) + ∂ θ (II( c ′ , h )) , (2.4)Since II and its derivatives are bounded on the compact set K , we have | ∂ θ ι ∗ h | . | ( ∇ N ∂ θ ) h | + | c ′ ||∇ N ∂ θ h | + | c ′ || h | + | ∂ θ c ′ || h | + | c ′ || ∂ θ h | . | ( ∇ N ∂ θ ) h | + | c ′ ||∇ N ∂ θ h | + | c ′ | (1 + | c ′ | ) | h | + |∇ N ∂ θ c ′ || h | . where we used (2.3) when changing ∂ θ to ∇ N ∂ θ (applied to c ′ and h ). Since n = 2, we have that | c ′ | < C and k h k L ∞ ≤ C k h k H c for some C > | ∂ θ ι ∗ h | . | ( ∇ N ∂ θ ) h | + |∇ N ∂ θ h | + | h | + k h k H c |∇ N ∂ θ c ′ | . Squaring and integrating, and using that k∇ N ∂ θ c ′ k L < C , we obtain that, k ∂ θ ι ∗ h k L . k ( ∇ N ∂ θ ) h k L + k∇ N ∂ θ h k L + k h k L + k h k H c . k h k H c , and therefore k h k H ( ι ) . k h k H c . The converse inequality follows in a similar manner, by using (2.4), to bound | ( ∇ N ∂ θ ) h | with | ∂ θ ι ∗ h | and lower order terms.For n > ∂ nθ ι ∗ h in terms of ( ∇ N ∂ θ ) n h and lower order terms that involve the secondfundamental form and its derivatives (as in (2.4)), and bounding the lowerorder terms in a similar manner. (cid:3) Reparametrization invariant Sobolev metrics on spaces ofcurves
The metric and geodesic equation in the smooth category.
Asdetailed in the introduction, we are interested in reparametrization invariant
OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 11
Sobolev metrics on the above defined spaces Imm( D, N ) and I n ( D, N ), and,more accurately, in metrics of the type (1.2): G c ( h, k ) = n X i =0 a i ( ℓ c ) Z D g ( ∇ i∂ s h, ∇ i∂ s k ) d s,a i ∈ C ∞ ((0 , ∞ ) , [0 , ∞ )) , for i = 0 , . . . , n and a , a n > , We now calculate the geodesic equation associated with G c in smoothsettings; in the next subsection we extend the treatment to Sobolev settings.To derive the geodesic equation it will be more convenient to write the metricusing the so-called inertia operator, i.e., use integration by parts to write G as(3.1) G c ( h, k ) = Z D g ( A c h, k ) d s + B c ( h, k ) . Here(3.2) A c : T c Imm( D, N ) → T c Imm( D, N ) , is called the inertia operator of the metric G and B c ( h, k ) depends solely onthe boundary of D and stems from the integration by parts process. Thusfor closed curves the operator B is not present. Lemma 3.1.
The inertia operator of the metric (1.2) takes the form: (3.3) A c ( h ) = n X i =0 ( − i a i ( ℓ c ) ∇ i∂ s h, For open curves, i.e. D = [0 , π ] , the boundary operator B is given by: (3.4) B c ( h, k ) = n X i =1 a i ( ℓ c ) i − X j =0 ( − i + j − g ( ∇ i + j∂ s h, ∇ i − j − ∂ s k ) (cid:12)(cid:12)(cid:12) π . Proof.
These formulas follow directly from the integration by parts formula(3.5) Z D g ( h, ∇ ∂ s k ) d s = g ( h, k ) | ∂D − Z D g ( ∇ ∂ s h, k ) d s . Note that for closed curves we have D = S and thus ∂D = ∅ . (cid:3) Before we calculate the geodesic equation we will collect variational for-mulas of several quantities that appear in the metric. In the following wewill denote the variation of a quantity in direction h ∈ T c Imm( D, N ) by D c,h . Lemma 3.2.
Let c ∈ Imm( D, N ) and h ∈ T c Imm( D, N ) . Then D c,h | c ′ | = g ( v, ∇ ∂ s h ) | c ′ | (3.6) D c,h d s = g ( v, ∇ ∂ s h ) d s (3.7) D c,h ℓ c = Z D g ( v, ∇ ∂ s h ) d s (3.8) where v = c ′ / | c ′ | denotes the unit length tangent vector to the curve c . Ex-tending the connection, as described in [9, Section 3] , we can also calcu-late the variation of the covariant derivtive ∇ ∂ s applied to a tangent vector k ∈ T c Imm( D, N ) : ∇ h ∇ ∂ s k = − g ( v, ∇ ∂ s h ) ∇ ∂ s k + ∇ ∂ s ∇ h k + R ( v, h ) k ;(3.9) Proof.
The first three formulas follow by straight-forward calculations, sim-ilar as for curves with values in Euclidean spaces, see, e.g., [28, 12]. For thelast formula we follow the more general presentation in [9], where the varia-tion of the Laplacian for D being a compact manifold of arbitrary dimensionhas been derived. Using the formula ∇ h ∇ ∂ θ k = ∇ ∂ θ ∇ h k + R ( h, c ′ ) k (3.10)for swapping covariant derivatives, see e.g. [9, Section 3.8], we obtain ∇ h ∇ ∂ s k = D f,h (cid:0) | c ′ | − (cid:1) ∇ ∂ θ k + | c ′ | − ∇ h ∇ ∂ θ k = − g ( ∇ ∂ s h, v ) ∇ ∂ s k + | c ′ | − ∇ ∂ θ ∇ h k + | c ′ | − R ( h, c ′ ) k which concludes the proof since v = | c ′ | − c ′ . (cid:3) We are now able to calculate the geodesic equation. In the following cal-culation we will restrict to first order metrics, for which the exact form of thegeodesic spray will be of importance in the proof of the local well-posednessresult. For higher order metrics the existence and well-posedness of the ge-odesic equation will follow from general principles on strong metrics and wewill thus not include these cumbersome calculations. The interested readercan consult the related calculations in [9], where the geodesic equations arederived for general higher order metrics (under the assumption that D hasno boundary). The geodesic equation for constant coefficient metrics onclosed curves in Euclidean space also appears in [14, Theorem 1.1]. Lemma 3.3.
The geodesic equation of the first order Sobolev type metric,as defined in (1.2) for n = 1 , is given by the set of equations: ∇ ∂ t ( A c c t ) = − g ( v, ∇ ∂ s c t ) A c c t −
12 Ψ c ( c t , c t ) ∇ ∂ s v − g ( ∇ ∂ s c t , A c c t ) v + a ( ℓ c ) R ( c t , ∇ ∂ s c t ) v, where the quadratic form Ψ c ( c t , c t ) is given by Ψ c ( c t , c t ) = a ( ℓ c ) g ( c t , c t ) + a ′ ( ℓ c ) Z D g ( c t , c t ) d s − a ( ℓ c ) g ( ∇ ∂ s c t , ∇ ∂ s c t ) + a ′ ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s For open curves, D = [0 , π ] , we get the following boundary conditions: (cid:16) − ∇ ∂ t ( a ( ℓ c ) ∇ ∂ s c t ) + Ψ c ( c t , c t ) v (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 , π = 0 . The proof of this result is postponed to Appendix A.1.
OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 13
The induced metric on Sobolev immersions.
To obtain the de-sired completeness and well-posedness results we consider the extension ofthe metric G (of order n ) on the Banach manifolds I q ( D, N ) ⊃ Imm( D, N ),for q ≥ max { n, } , as defined in Definition 2.1 above.Our aim in the rest of the section is to show the smoothness of the metrics G on I q ( D, N ) (assuming q ≥ n ). First, we need to introduce some mixedorder spaces: Definition 3.4.
Let q ≥ q ≥ k ≥
0. We define the function space: H k I q ( D, T N ) = n h ∈ H k ( D, T N ) : π N ◦ h ∈ I q ( D, N ) o . We have the following result concerning their manifold structure and theoperator ∇ ∂ θ : Lemma 3.5.
The spaces H k I q ( D, T N ) are smooth Hilbert manifolds for any q ≥ and q ≥ k ≥ . The mapping (3.11) ∇ ∂ θ : H k I q ( D, T N ) → H k − I q ( D, T N ) is a bounded linear mapping for ≤ k ≤ q .Proof. The first part of this result can be found in [10, Theorem 2.4], whilethe second part follows directly from the definition of the space H k I q ( D, T N ). (cid:3) Note that H q I q ( D, T N ) = T I q ( D, N ). If k < q then H k I q ( D, T N ) is thethe robust fiber completion of the weak Riemannian manifold ( I q ( D, T N ) , G k )with the Sobolev metric G k from (1.2) in the sense described in [26]. Thesespaces will appear, when we repeatedly apply ∇ ∂ s to a vector field h alongan H n -immersion ( ∇ ∂ s will reduce the order of the vector field, but not ofits foot point). To show the smoothness of the metric we need the followingresult: Lemma 3.6.
Let q ≥ . Then the mapping H k +1 I q ( D, T N ) → H k I q ( D, T N )(3.12) h
7→ ∇ ∂ s h = 1 | π ( h ) | ∇ ∂ θ h (3.13) is smooth for any k ≥ .Proof. The mapping H k +1 I q ( D, T N ) → H k I q ( D, T N )(3.14) h
7→ ∇ ∂ θ h (3.15)is smooth by Lemma 3.5. By the module properties of Sobolev spaces mul-tiplication H q ( D, R ) × H k I q ( D, T N ) → H k I q ( D, T N ) is smooth for q ≥ k ≥
0. Thus the result follows since | π ( h ) | ∈ H q ( D, R ). (cid:3) Using this lemma we immediately obtain the smoothness of the metric:
Theorem 3.7.
Let q ≥ . Consider the Sobolev metric G on Imm( D, N ) of order n ≤ q of the form (1.2) . Then G extends to a smooth Riemannianmetric on I q ( D, N ) . For q = n the metric G is a strong Riemannian metricon I n ( D, N ) . Proof.
Iterating Lemma 3.6 we have that(3.16) ∇ i∂ s : T I q ( D, N ) = H q I q ( D, T N ) → H q − i I q ( D, T N ) ⊂ L I q ( D, T N )is smooth for 0 ≤ i ≤ n . Thus the mapping T I q ( D, N ) × I q T I q ( D, N ) → L ( D, R )( h, k ) g c ( ∇ i∂ s h, ∇ i∂ s k ) | c ′ | is smooth as well. Here we used again the module properties of Sobolevspaces. It remains to show the smoothness of c ℓ c . Therefore we use thefact that L ( D, R ) → R f Z f d θ is a bounded linear operator, hence it immediately follows that the lengthfunction c ℓ c = R | c ′ | d θ is smooth. For n ≥ G is a strongRiemannian metric on I n ( D, N ) since for each c ∈ I n ( D, N ) the innerproduct G c ( h, k ) describes the Hilbert space structure on T c I n ( D, N ) (Thisis best seen in a local chart, whose base is, by definition, around a smooth c ,otherwise one has to deal with Γ H n ( c ∗ T N ) for c a Sobolev H n -immersion). (cid:3) Local well-posedness of the geodesic equation.
The local well-posedness results as summarized in the following theorem are based on theseminal method of Ebin and Marsden [20]. They are known in the caseof closed curves, see [10, 28, 6], but to the best of our knowledge they arenew for the case of open curves. However, as local well-posedness is notthe focus of the current article, we postpone the proof of this result to theAppendix A.2.
Theorem 3.8.
Let D = [0 , π ] or D = S . Let G be a Sobolev metric oforder n ≥ of the form (1.2) on I q ( D, N ) , with either q ≥ n or q = n ≥ .We have: (1) The initial value problem for the geodesic equation has unique localsolutions in the Banach manifold I q ( D, N ) . The solutions dependsmoothly on t and on the initial conditions c (0 , · ) and c t (0 , · ) . More-over, the Riemannian exponential mapping exp exists and is smoothon a neighborhood of the zero section in the tangent bundle, and ( π, exp) is a diffeomorphism from a (possibly smaller) neighborhoodof the zero section of T I q ( D, N ) to a neighborhood of the diagonalin I q ( D, N ) × I q ( D, N ) . (2) The results of part 1 (local well-posedness of the geodesic equationand properties of the exponential map) continue to hold on I q ( D, N ) ∩ C ∞ loc ( D o , N ) , where D o is the interior of D . Note, that for D = S we have Imm( S , N ) = I q ( S , N ) ∩ C ∞ loc ( S , N ),i.e., the local well-posedness continues to hold in the smooth category. OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 15 Estimates
In this section we prove some interpolation inequalities for Sobolev sec-tions of the tangent bundle, that will be needed for proving metric complete-ness of ( I n ( D, N ) , G ) in various cases. For vector-space-valued functions,these inequalities are rather simple adaptations of standard inequalities; thisis the case when N = R d , as sections of c ∗ T R d can be regarded as vector-space-valued functions (see [14, Lemmas 2.14–2.15], [12, Lemma 2.4] for thecase D = S ).For a general target manifold, two things change: first, instead of workingwith a section h ∈ H k ( D, c ∗ T N ) directly, we need to parallel transport h toa single base point, that is, to work with H ( θ ) = Π θ θ h ( θ ) ∈ H k ( D, T c ( θ ) N ) ≃ H k ( D, R dim N ) , where θ ∈ D is a base point, and Π θ θ is the parallel transport, in N ,from T c ( θ ) N to T c ( θ ) N , along c . The reason for using H is that it is avector-space-valued function, and so we can take regular derivatives of H and use the fundamental theorem of calculus. The derivatives of H relateto covariant derivatives of h via(4.1) H ′ ( θ ) = d d θ Π θ θ h ( θ ) = Π θ θ ∇ ∂ θ h ( θ ) . See, e.g., [17, Chapter 2, exercise 2]. Note that, since the parallel transportoperator is an isometry, we have | H ( θ ) | = | h ( θ ) | , | H ′ ( θ ) | = |∇ ∂ θ h ( θ ) | , andso on for higher order derivatives.The second difference from the Euclidean case arises when D = S . Inthe Euclidean case we obtain inequalities for periodic functions, that aregenerally better than the ones for general functions (and this fact is essentialfor completeness of constant coefficients metrics). However, when N 6 = R d ,even though h (0) = h (2 π ), it is not true that H (0) = H (2 π ), because theholonomy along the curve c is in general non-trivial (that is, Π π = id T c (0) N ).Therefore, we need to bound the amount by which H fails to be periodic,and to prove estimates for such ”almost periodic” functions.We now state the estimates; first the inequalities that hold for both D = S or D = [0 , π ], and then inequalities that hold only in the periodic case.As the proof of the periodic case is long and somewhat different from therest of the analysis in this paper, we postpone it to Appendix B. This is donesolely for the sake of readability — these estimates are at the core of provingthe metric completeness of ( I n ( S ; N ); G ) for G with constant coefficients,and are one of the main differences between the analysis of manifold valuedcurves and of R d -valued curves. Lemma 4.1 (General estimates) . If n ≥ , c ∈ I n ( D, N ) and h ∈ H n ( D, c ∗ T N ) ,then for ≤ k < n , there exists C = C ( k, n, dim N ) > such that (4.2) a k k∇ k∂ s h k L ( d s ) ≤ C (cid:16) k h k L ( d s ) + a n k∇ n∂ s h k L ( d s ) (cid:17) , and (4.3) a k k∇ k∂ s h k L ∞ ≤ C (cid:16) a − k h k L ( d s ) + a n − k∇ n∂ s h k L ( d s ) (cid:17) , for every a ∈ (0 , ℓ c ] . The same holds when we replace ∇ ∂ s with ∇ ∂ θ and d s with d θ , with a ∈ (0 , π ] .Proof. Since all the norms involved (in the d s case) are reparametrization-invariant, we can assume that c is arc-length parametrized. In this case, wehave ∇ ∂ s = ∇ ∂ θ , d s = d θ , where θ ∈ [0 , ℓ c ] (and in the case D = S , weidentify the points θ = 0 and θ = ℓ c ). Define H : [0 , ℓ c ] → T c (0) N ≡ R dim N H ( θ ) = Π θ h ( θ ) . From (4.1) we have (cid:12)(cid:12)(cid:12) ∇ k∂ θ h ( θ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Π θ ∇ k∂ θ h ( θ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) d k d θ k H ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) . In order to prove (4.2), we therefore need to prove that a k Z ℓ c | ∂ kθ H | d θ ≤ C (cid:18)Z ℓ c | H | d θ + a n Z ℓ c | ∂ nθ H | d θ (cid:19) , for every a ∈ (0 , ℓ c ], and similarly for (4.3). Since H is valued in R dim N ,this is a standard Sobolev inequality, see, e.g., [25, Theorem 7.40].The d θ case is similar, but simpler (no need to reparametrize c first). (cid:3) Lemma 4.2 (Estimates for S ) . If n ≥ , c ∈ I n ( S , N ) and h ∈ H n ( S , c ∗ T N ) ,then for < k < n , there exists C > , depending on k, n, dim N , the injec-tivity radius and the upper and lower bounds on the sectional curvature of N , such that (4.4) k∇ k∂ s h k L ( d s ) ≤ C min (cid:8) , ℓ c (cid:9) (cid:16) k h k L ( d s ) + k∇ n∂ s h k L ( d s ) (cid:17) . Proof.
See Appendix B. (cid:3)
Remark 4.3.
It is interesting to compare inequality (4.4) to the equivalentone in the Euclidean settings [14, Lemma 2.14], that is, when N = R d .There we have k∇ ∂ s h k L ( d s ) ≤ ℓ c k∇ ∂ s h k L ( d s ) , from which higher order inequalities readily follow. The zeroth order termthat appears in the right-hand side of (4.4) is a curvature term, and, as theproof in Appendix B shows, arise from the non-trivial holonomy along theclosed curve c . 5. Metric and geodesic completeness
We now want to prove the main result of this article, i.e., extend thecompleteness results, obtained for planar curves, to the situation studiedin this article. The exact statement of the main results is now detailed inTheorems 5.1–5.3 below (the main result as presented in the introduction isa slightly simplified form of them).
Theorem 5.1.
Let n ≥ , let D = [0 , π ] or D = S , and let G be asmooth Riemannian metric on I n ( D, N ) . Assume that for every metric ball B ( c , r ) ∈ ( I n ( D, N ) , dist G ) , there exists a constant C = C ( c , r ) > , suchthat k h k G c ≥ Cℓ − / c k∇ ∂ s h k L ( d s ) , (5.1) OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 17 k h k G c ≥ C k∇ k∂ s h k L ∞ k = 0 , . . . , n − , (5.2) k h k G c ≥ C k∇ n∂ s h k L ( d s ) . (5.3) Then G is a strong metric, and we have: (1) ( I n ( D, N ) , dist G ) is a complete metric space. (2) ( I n ( D, N ) , G ) is geodesically complete For Sobolev metrics the type (1.2) we also obtain geodesic convexity:
Theorem 5.2.
Let D = [0 , π ] or D = S , and let G be a smooth Sobolevmetric of the type (1.2) on I n ( D, N ) , that satisfies assumptions (5.1) – (5.3) .Then any two immersions in the same connected component can be joinedby a minimizing geodesic. The reason that in Theorem 5.2 we further assume, unlike in Theorem 5.1,that G is of the type (1.2) is merely a technical one; both theorems are firstproved for metrics of this type, and in Theorem 5.1 the extension to thegeneral case is immediate. Theorem 5.2, with the same method of proof,definitely holds for metrics that are not of type (1.2), but this needs tobe checked on a case-by-case basis, and thus we present this theorem onlyfor these type of metrics. The assumptions (5.1)–(5.3) are satisfied in thefollowing cases: Theorem 5.3.
Let D = [0 , π ] or D = S , and let G be a Sobolev metric oforder n ≥ of the type (1.2) on I n ( D, N ) . Assume that one of the followingholds: (1) Length weighted case:
There exists α > such that either a ( x ) ≥ αx − or both a ( x ) ≥ αx − and a k ( x ) ≥ αx k − for some k > . (2) Constant coefficient case: D = S and both a and a n are posi-tive constants.Then assumptions (5.1) – (5.3) hold, and the completeness results of Theo-rem 5.1 hold for ( I n ( D, N ) , G ) . Remark 5.4.
Note that the family of scale-invariant Sobolev metric, as in-troduced in Section 1.2, satisfies conditions (1) of Theorem 5.3. In the arti-cle [16], where the authors study completeness properties for length weightedmetrics on curves with values in Euclidean space, more general conditionson the coefficient functions that still ensure completeness have been derived.While such an analysis should be also possible in our situation, the resultingconditions would be much more complicated. The reason for this essentiallylies in the fact that the manifold valued Sobolev estimates are more compli-cated (and involve lower-order terms), compared to the R d -valued one, asdescribed in Remark 4.3. Thus, for the sake of clarity, we discuss here onlyconditions of the type (1).The remaining part of this section will contain the proof of these theorems.To prove Theorem 5.1 we will first show the metric completeness, which thenimplies the geodesic completeness, see [24, VIII, Proposition 6.5]. Sincethe theorem of Hopf-Rinow is not valid in infinite dimensions we cannot Atkin constructed in [3] an example of a geodesically complete Riemannian manifoldwhere the exponential map is not surjective, see also [21]. conclude the existence of geodesics by abstract arguments. Instead we showthis statement by hand using the direct methods of the calculus of variations,in Section 5.5. Finally, in Section 5.6, we deduce geodesic completeness inthe smooth category.5.1.
Reduction from metric completeness to equivalence of strongRiemannian metrics.
In this section we reduce the question of metriccompleteness ( I n ( D, N ) , dist G ) to a question on uniform equivalence of theRiemannian metrics G and H on metric balls. This is done in two steps. First reduction — distance equivalence on balls.
The space I n ( D, N )is an open subset of H n ( D, N ) (see Proposition 2.2). In addition to the met-ric dist G induced by G , it therefore inherits also the distance dist H functioninduced from H n ( D, N ). In general, dist G and dist H are not equivalent.However, we do have the following: Proposition 5.5.
Assume that G is a strong Riemannian metric on I n ( D, N ) and (1) For every metric ball B ( c , r ) ⊂ (cid:0) I n ( D, N ) , dist G (cid:1) , there exists aconstant C > such that dist H ≤ C dist G on B ( c , r ) . (2) For every metric ball B ( c , r ) ⊂ (cid:0) I n ( D, N ) , dist G (cid:1) , k c ′− k L ∞ isbounded.Then ( I n ( D, N ) , dist G ) is metrically complete.Proof. The proof below is similar to the proof of [12, Theorem 4.3]. For theconvenience of the reader we repeat the arguments here. Given a Cauchy se-quence ( c n ) in (cid:0) I n ( D, N ) , dist G (cid:1) , the sequence remains in a bounded metricball in (cid:0) I n ( D, N ) , dist G (cid:1) , hence by (1) the sequence is also a Cauchy se-quence in H n ( D, N ), hence c n → c ∈ H n ( D, N ) (modulo a subsequence).Moreover, since the sequence c n lies in a metric ball, | ( c ′ n ) − | < C < ∞ for all n by (2), and since H n convergence implies C convergence, we ob-tain that | c ′− | ≤ C , and thus c ∈ I n ( D, N ). Since both H and G arestrong metrics on I n ( D, N ), they induce the same topology (the manifoldtopology) [24, VII, Proposition 6.1], and thus dist H ( c n , c ) → G ( c n , c ) →
0, hence (cid:0) I n ( D, N ) , dist G (cid:1) is metrically complete. (cid:3) Second reduction — metric equivalence implies distance equiv-alence.
Next, we show that distance-equivalence on metric balls follows frommetric-equivalence on metric balls. The following proposition is the con-tent of Proposition 3.5 and Remark 3.6 in [12], adapted to our setting.
Proposition 5.6.
Assume that for each metric ball B ( c , r ) ⊂ (cid:0) I n ( D, N ) , dist G (cid:1) , there exists C = C ( c , r ) > such that for every c ∈ B ( c , r ) and h ∈ H n ( D ; c ∗ T N ) , we have (5.4) k h k H c ≤ C k h k G c . Then, property (1) in Proposition 5.5 holds.
OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 19
Proof.
The following proof is an adaptation of the proof of [12, Lemma 4.2].Let c , c ∈ B ( c , r ) and ε >
0, and let γ be a piecewise smooth curvebetween c and c with L G ( γ ) ≤ dist G ( c , c ) + ε . Since dist G ( c , c ) < r ,by the triangle inequality, we have that γ ⊂ B ( c , r ). We then have, usingassumption (5.4) for B ( c , r ), thatdist H ( c , c ) ≤ L H ( γ ) ≤ CL G ( γ ) ≤ C (dist G ( c , c ) + ε ) . Since ε is arbitrary, completes the proof. (cid:3) Estimates on ℓ c and | c ′ | in metric balls. In this section we boundvarious quantities that depend on the curve c uniformly on metric balls in I n ( D, N ). These will enable us to prove the assumption of Proposition 5.6,as well as assumption (2) of Proposition 5.5.To this end, we will repeatedly use the following result (see [12, Lemma 3.2]for a proof ): Lemma 5.7.
Let ( M , g ) be a Riemannian manifold, possibly of infinitedimension, and let F be a normed space. Let f : M → F be a C -function,such that for each metric ball B ( y, r ) in M there exists a constant C , suchthat k T x f.v k F ≤ C (1 + k f ( x ) k F ) k v k x for all x ∈ B ( y, r ) , v ∈ T x M . Then f is Lipschitz continuous on every metric ball, and in particular boundedon every metric ball. Moreover, if the constant C is independent of the met-ric ball B ( y, r ) , then the Lipschitz constant in B ( y, r ) can be bounded by afunction L : [0 , ∞ ) → (0 , ∞ ) , increasing in all variables, as follows: k f ( x ) − f ( x ) k F ≤ L ( C, k f ( y ) k F , r ) dist( x , x ) for every x , x ∈ B ( y, r ) . In particular the Lipschitz constant in B ( y, r ) depends on y only through k f ( y ) k F . Remark 5.8.
Tracking the constants in Lemma 5.7 carefully, one can obtainthe bound(5.5) L ( C, t, r ) = C (1 + r )(1 + 2 r ) e Cr (1 + t ) . Note that this is not sharp, it is simply what is obtained by the method ofthe proof (using Gronwall’s inequality).
Lemma 5.9 (Bounds on length) . Assume that assumption (5.1) holds.Then the length function c ℓ c is bounded from above and away fromzero on every metric ball.Proof. From (3.8) we have that | D c,h ℓ c | ≤ Z D | g ( v, ∇ ∂ s h ) | d s ≤ ℓ / c (cid:18)Z D | g ( v, ∇ ∂ s h ) | d s (cid:19) / ≤≤ ℓ / c k∇ ∂ s h k L ( d s ) . In fact, for the case in which C is independent of the metric ball, the statement in[12, Lemma 3.2] is inaccurate; the statement of Lemma 5.7 is the corrected one, and theproof follows exactly as in [12, Lemma 3.2], by checking carefully which constants appearwhen using Gronwall’s inequality [12, Corollary 2.7]. Therefore, under the assumption (5.1), we have | D c,h ℓ c | . k h k G c and we obtain from Lemma 5.7 that c ℓ c is bounded on every metric ball.Similarly, for the map c ℓ − c , we have, under the assumption (5.1), that | D c,h ℓ − c | = ℓ − c | D c,h ℓ c | ≤ ℓ − / c k∇ ∂ s h k L ( d s ) . ℓ − c k h k G c , which concludes the proof using again Lemma 5.7. (cid:3) Lemma 5.10 (Bounds on speed) . Assume that assumption (5.2) holds for k = 1 . Then, there exists a constant α = α ( c , r ) > such that α − ≤ | c ′ ( θ ) | ≤ α for every c ∈ B ( c , r ) and θ ∈ D .Proof. Consider the functionlog | c ′ | : ( I n ( D, N ) , G ) → L ∞ ( D ; R ) . By (3.6) and assumption (5.2) we have k D c,h log | c ′ |k L ∞ ≤ k g ( v, ∇ ∂ s h ) k L ∞ ≤ k∇ ∂ s h k L ∞ . k h k G c . By Lemma 5.7 we thus have that log | c ′ | is bounded on metric balls, fromwhich the claim follows. (cid:3) Lemma 5.11.
Assume that assumption (5.2) holds for k = 0 . Then theimage in N of every metric ball B ( c , r ) is bounded. That is, there exists R = R ( c , r ) > such that for every c ∈ B ( c , r ) and every θ ∈ D , dist N ( c ( θ ) , c (0)) < R. Proof.
Let c ∈ B ( c , r ), and let c ( t, θ ) : [0 , → B ( c , r ) be a path between c = c (0 , · ) to c = c (1 , · ), whose length is smaller than r . Using (5.2), wehavedist N ( c ( θ ) , c ( θ )) ≤ Z | ∂ t c ( t, θ ) | d t ≤ C Z k ∂ t c ( t, θ ) k G c d t < Cr. This completes the proof, as the length of c is finite. (cid:3) Lemma 5.12.
Assume that assumptions (5.1) – (5.3) hold. Then the follow-ing quantities are bounded on every metric ball k∇ k∂ s | c ′ |k L ∞ k = 0 , . . . , n − , (5.6) k∇ k∂ s | c ′ |k L k = 0 , . . . , n − , (5.7) where L is with respect to either d s or d θ .Proof. The proof of this is result follows by an induction on k using itera-tively Lemma 5.9 and 5.10. It is mainly an adaptation of Lemma 3.3 andProposition 3.4 in [12], though the calculations in our situation are moreinvolved due to the appearance of curvature terms of the manifold N . Tokeep the presentation simple we postpone it to the Appendix C. (cid:3) OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 21
Proof of Theorem 5.1: metric and geodesic completeness.
Weare now able to prove Theorem 5.1, that is, that ( I n ( D, N ) , G ) is metricallyand geodesically complete. We first prove it for a metric G of the type (1.2)of order n that satisfies assumptions (5.1)–(5.3). Afterwards the assumptionthat G is of the type (1.2) will be removed.In particular, G satisfies (5.2), and therefore Lemma 5.10 implies thatassumption (2) in Proposition 5.5 holds. Therefore, in order to prove that( I n ( D, N ) , dist G ) is metrically complete, we need to show that G is a strongmetric, and prove property (1), which by Proposition 5.6 follows from (5.4).In fact, we will show a stronger result and prove that G and H are equivalentuniformly on metric balls. This will also imply that G is a strong metric.From Lemma 4.1, we have n X i =0 k∇ i∂ θ h k L ( d θ ) ≤ C k h k H for some universal constant C >
0. Similarly, we have(5.8) k∇ ∂ θ h k L ∞ ≤ C ′ k h k H for some universal constant C ′ > ∇ ∂ s we have, by using the Leibniz rule, ∇ k∂ s h = 1 | c ′ | k ∇ k∂ θ h + k − X i =1 P i,k ∇ i∂ θ h, where P i,k are polynomials in | c ′ | , ∇ ∂ s | c ′ | , . . . , ∇ k − i∂ s | c ′ | and | c ′ | − , . . . , | c ′ | − k ,which are linear in ∇ k − i∂ s | c ′ | . Similarly, ∇ k∂ θ h = | c ′ | k ∇ k∂ s h + k − X i =1 Q i,k ∇ i∂ s h where Q i,k is a polynomial in the variables | c ′ | , ∇ ∂ s | c ′ | , . . . , ∇ k − i∂ s | c ′ | and thevariables | c ′ | , . . . , | c ′ | k − , which are linear in ∇ k − i∂ s | c ′ | . Using Lemma 5.10 andLemma 5.12, we therefore have that for k < n , P i,k and Q i,k are uniformlybounded on any metric ball, and so are | c ′ | ± , hence |∇ k∂ s h | . k X i =1 |∇ i∂θ h | , |∇ k∂θ h | . k X i =1 |∇ ik h | , uniformly on every metric ball. The bound on | c ′ | ± also implies that inte-gration with respect to d s or d θ are equivalent, hence(5.9) k∇ k∂ s h k L ( d s ) . k X i =1 k∇ i∂θ h k L ( d θ ) , k∇ k∂θ h k L ( d θ ) . k X i =1 k∇ ik h k L ( d s ) , uniformly on every metric ball.For k = n , we have, uniformly on every metric ball, |∇ n∂ s h | . (cid:12)(cid:12) ∇ n − ∂ s | c ′ | (cid:12)(cid:12) |∇ ∂ θ h | + n X i =2 |∇ i∂ θ h | and |∇ n∂θ h | . (cid:12)(cid:12) ∇ n − ∂ s | c ′ | (cid:12)(cid:12) |∇ ∂ s h | + n X i =2 |∇ i∂ s h | , and therefore, invoking Lemma 5.12 again and using (5.8), we have,(5.10) k∇ n∂ s h k L ( d s ) . k∇ ∂ θ h k L ∞ + n X i =2 k∇ i∂ θ h k L ( d θ ) . C k h k H and, using (5.2) again,(5.11) k∇ n∂θ h k L ( d θ ) . k∇ s h k L ∞ + n X i =2 k∇ i∂ s h k L ( d s ) . k h k G c + n X i =2 k∇ i∂ s h k L ( d s ) . Since (5.1) holds, we have by Lemma 5.9 that ℓ c is uniformly boundedfrom above and below on metric balls, hence all the coefficient functions a i ( ℓ c ) ≥ a , a n are alsobounded away from zero. We therefore have that, on each metric ball k h k L ( d s ) + k∇ n∂ s h k L ( d s ) . k h k G c . n X i =0 k∇ i∂ s h k L ( d s ) . Since ℓ c is bounded from below and above uniformly on metric balls, Lemma 4.2enables us to improve that to n X i =0 k∇ i∂ s h k L ( d s ) . k h k G c . n X i =0 k∇ i∂ s h k L ( d s ) Combining this with the estimate (5.9), (5.10) and (5.11) immediately imply k h k H c . k h k G . k h k H c , uniformly on metric balls. In particular, this implies (5.4) and show that G isa strong metric, thus all the assumptions of Propositions 5.5–5.6 are satisfied,which completes the proof of metric completeness. As stated before, geodesiccompleteness follows directly as for strong Riemannian metrics (in infinitedimensions) metric completeness still implies geodesic completeness, see,e.g., [24, VIII, Proposition 6.5].We now remove the assumption that G is of the type (1.2), and onlyassume that it is a smooth metric that satisfies (5.1)–(5.3). Denote by ˜ G the metric k h k G c := k h k L ( d s ) + ℓ − c k∇ ∂ s h k L ( d s ) + k∇ n∂ s h k L ( d s ) . This metric is of the type (1.2), and in Section 5.4 below we show that thismetric indeed satisfies (5.1)–(5.3). Therefore, it is metrically complete.Now assume that G is another metric that satisfies (5.1)–(5.3). We claimthat on every metric ball B G ( c , r ), there exists a constant C = C ( c , r )such that k · k ˜ G c ≤ C k · k G c . Indeed, assumptions (5.1) and (5.3) implythat G controls the second and third addends in the definition on ˜ G ; since k h k L ∞ ≥ ℓ − / c k h k L ( d s ) , assumption (5.2) for k = 0 and Lemma 5.9 implythat G controls the second addend in ˜ G as well (uniformly on every metricball). This implies, in particular, that G is a strong metric (since ˜ G is). OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 23
The proof is now concluded by similar arguments as Section 5.1 (with ˜ G instead of H ): Let c k ∈ ( I n ( D, N ) , dist G ) be a Cauchy sequence. It followsthat c k is also a Cauchy sequence in ( I n ( D, N ) , dist ˜ G ). Since ( I n ( D, N ) , dist ˜ G )is metrically complete, c k converges in ( I n ( D, N ) , dist ˜ G ) to some limit c ∈I n ( D, N ). Since both G and ˜ G are strong metrics on I n ( D, N ), they in-duce the same topology. Therefore, c k → c in ( I n ( D, N ) , dist G ) as well, thusproving metric completeness, from which geodesic completeness follows asbefore.5.4. Proof of Theorem 5.3. Length weighted case.
If both a ( x ) ≥ αx − and a k ( x ) ≥ αx k − for some k >
1, then by (4.2) we have that ℓ − c k∇ ∂ s h k L ( d s ) ≤ C k h k G for some C >
0. This is also obviously true if a ( x ) ≥ αx − . Thus (5.1)holds, and from Lemma 5.9 we obtain that the length function c ℓ c isbounded from above and away from zero on any metric ball. Since G is ofthe type (1.2), we have that k h k G c ≥ a ( ℓ c ) k h k L ( d s ) + a n ( ℓ c ) k∇ n∂ s h k L ( d s ) , and the bound on the length implies that on each metric ball, the constants a ( ℓ c ) and a n ( ℓ c ) are bounded away from zero. This immediately implies(5.3), and also that on every metric ball k h k G c ≥ C ( k h k L ( d s ) + k∇ n∂ s h k L ( d s ) ) , for some C >
0. On the other hand, using (4.3) with a = ℓ c we have, forevery k = 0 , . . . , n − k∇ k∂ s h k L ∞ ≤ C (cid:16) ℓ − k − c k h k L ( d s ) + ℓ n − k ) − c k∇ n∂ s h k L ( d s ) (cid:17) , hence on each metric ball, we have k∇ k∂ s h k L ∞ ≤ C ′ (cid:16) k h k L ( d s ) + k∇ n∂ s h k L ( d s ) (cid:17) ≤ C ′′ k h k G c , which implies (5.2). Constant coefficient case.
Assume that a and a n are positive con-stants. We then immediately have (5.3). Furthermore, using (4.4) for k = 1,we have k∇ ∂ s h k L ( d s ) ≤ Cℓ c k h k G c for some constant C that is independent of the curve c . This implies (5.1),and hence the boundedness of c ℓ c by Lemma 5.9. The proof of (5.2)now follows in the same manner as the length weighted case. This is the crucial point in which the improved estimates for closed curves inLemma 4.2 are needed.
Proof of Theorem 5.2: existence of minimizing geodesics.
Wenow prove that any two immersions in the same connected component canbe joined by a minimizing geodesic. The approach is a variational one: weconsider the energy E ( c ) := Z G c ( ˙ c, ˙ c ) d t, defined on the set A c ,c := n c : [0 , → I n ( D, N ) : ˙ c ∈ L ((0 , H n ( D ; c ∗ T N )) ,c (0) = c , c (1) = c o , where c , c ∈ I n ( D, N ) are two immersions in the same connected compo-nent (thus A c ,c is a non-empty set). We aim to show that there exists aminimizer to E over A c ,c , which is, by definition, a minimizing geodesic.We prove the existence of minimizers using the direct methods in thecalculus of variations; namely, we take a minimizing sequence c j , provethat it is weakly sequentially precompact, and that any limit point mustbe a minimizer. In order to use weak convergence, we embed the curvesin a Hilbert space, which neither I n ( D, N ) or H n ( D, N ) are (this is thepoint where N -valued curves differ from R d -valued curves treated in [12,Theorem 5.2]). To this end, we again isometrically embed N into R m forsome large enough m ∈ N , as in the definition of H n ( D, N ) that we startedwith (Definition 2.1). This will require us, as in Section 2, to use Lemma 2.3to relate the metric H on H n ( D, N ) with the standard Sobolev norm on H n ( D ; R m ).Let now c j ∈ A c ,c be a minimizing sequence of E , that is, E ( c j ) → inf A c ,c E. In particular, E ( c j ) is a bounded sequence. Denote by R its supremum.We also fix an isometric embedding ι : N → R m , and, using this embedding,we consider c j as elements of the Hilbert space H ([0 , H n ( D ; R m )). Step I: The family ( c j ( t )) j ∈ N ,t ∈ [0 , lies in a bounded ball around c . Fix t ∈ [0 ,
1] and j ∈ N . Since c j : [0 , t ] → I n ( D, N ) is a path from c to c j ( t ), we havedist G ( c j ( t ) , c ) ≤ (cid:18)Z t k ˙ c j ( t ) k G cj ( t ) d t (cid:19) ≤ Z k ˙ c j ( t ) k G cj ( t ) d t = E ( c j ) ≤ R . Therefore, ( c j ( t )) j ∈ N ,t ∈ [0 , ⊂ B ( c , R ), where the ball is with respect to themetric G . Step II: The family ( c j ) j ∈ N is a bounded set in H ([0 , H n ( D ; R m )) . Since G satisfies (5.1)–(5.3), we have that (5.4) hold uniformly on B ( c , R ),that is, there exists C > C − k h k H c ≤ k h k G c ≤ C k h k H c , for all c ∈ B ( c , R ) , h ∈ H n ( D ; c ∗ T N ) . This was proved in Section 5.3. Moreover, from Lemmata 5.11–5.12, wehave that the assumptions of Lemma 2.3 hold uniformly on B ( c , R ), hence, OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 25 combining with the above inequality, we obtain that there exists
C > C − k h k H n ( ι ) ≤ k h k G c ≤ C k h k H n ( ι ) , for all c ∈ B ( c , R ) , h ∈ H n ( D ; c ∗ T N ) . Since ( c j ( t )) j ∈ N ,t ∈ [0 , ⊂ B ( c , R ), we obtain that for any fixed t and j , k c − c j ( t ) k H n ( ι ) = Z D | c − c j ( t ) | + | ∂ nθ ( c − c j ( t )) | d θ = Z D (cid:12)(cid:12)(cid:12)(cid:12)Z t ˙ c j ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z t ∂ nθ ˙ c j ( t ) d t (cid:12)(cid:12)(cid:12)(cid:12) d θ ≤ Z D Z | ˙ c j ( t ) | + | ∂ nθ ˙ c j ( t ) | d t d θ = Z k ˙ c j ( t ) k H n ( ι ) d t = C Z k ˙ c j ( t ) k G cj ( t ) d t ≤ CR . And therefore k c j k H ([0 , H n ( D ; R m )) = Z k c j ( t ) k H n ( ι ) + k ˙ c j ( t ) k H n ( ι ) d t ≤ Z k c k H n ( ι ) + 2 k c − c j ( t ) k H n ( ι ) d t + Z k ˙ c j ( t ) k H n ( ι ) d t ≤ Z k c k H n ( ι ) + 2 CR d t + C Z k ˙ c j ( t ) k G cj ( t ) d t ≤ CR + 2 k c k H n ( ι ) Hence, the sequence c j is bounded in the Hilbert space H ([0 , H n ( D ; R m )).Therefore, it has a subsequence (not relabeled) that weakly converges tosome c ∗ ∈ H ([0 , H n ( D ; R m )). Step III: The limit point c ∗ belongs to A c ,c . Let ε ∈ (0 , / H ([0 , H n ( D ; R m )) ⊂ C ([0 , H n − ε ( D ; R m ))is compact (due to the Aubin–Lions–Simon Lemma ) and H n − ε ( D ; R m ) iscompactly embedded in C n − ( D ; R m ). In particular, we thus have that c j → c ∗ in the strong topology of C ([0 , C n − ( D ; R m )). Since c j ( θ ) ∈ N for all j and θ , the uniform convergence implies that c ∗ ( θ ) ∈ N for all θ aswell. Since c j (0) = c and c j (1) = c for all j , the same holds for c ∗ . Finally,since c j ( t ) ∈ B ( c , R ) for every j and t , Lemma 5.10 implies that | ∂ θ c j ( t, θ ) | > α for some α >
0. Since c j → c ∗ in C ([0 , C n − ( D ; R m )), the same holds for c ∗ , hence c ∗ ∈ I n ( D, N ). This shows that indeed c ∗ ∈ A c ,c . Step IV: Weak convergence of derivatives.
It will be helpful now toemphasize the particular curve that is used to define the ∇ ∂ s derivative. See, e.g., [11, Theorem II.5.16]. With the respect to the notation there we use thelemma for p = ∞ , r = 2, B = H n , B = H n − ε and B = H n − . We can use p = ∞ because H embeds in L ∞ . Therefore, for the rest of this proof, denote D c j := | c j | − ∇ N ∂θ . We now showthat, for k = 0 , . . . , n , we have(5.12) D nc j ˙ c j ⇀ D nc ∗ ˙ c ∗ in L ([0 , L ( D ; R m )) . By the definition of c ∗ , we have that˙ c j ⇀ ˙ c ∗ in L ([0 , H n ( D ; R m )) , hence the case k = 0 is immediate. We will show that for k = 1 , . . . , n ,(5.13) h j ⇀ h in L ([0 , H k ( D ; R m ))implies(5.14) D c j h j ⇀ D c ∗ h in L ([0 , H k − ( D ; R m )) , from which (5.12) follows by induction. First, considering all the vectorfields as sections of D × R m , we have that D c j h j = 1 | ∂ θ c j | (cid:0) ∂ θ h j − II c j ( ∂ θ c j , h j ) (cid:1) , D c ∗ h = 1 | ∂ θ c ∗ | ( ∂ θ h − II c ∗ ( ∂ θ c ∗ , h )) , where the subscript of II denotes the point where it is evaluated (recall thatII is the second fundamental form of N in R m ).Since c j → c ∗ in C ([0 , C n − ( D ; R m )) and | ∂ θ c j | is uniformly boundedfrom below, we have that | ∂ θ c j | − → | ∂ θ c ∗ | − uniformly (in t and θ ). Inparticular, since II c j are uniformly bounded bilinear forms (this followsagain from Lemma 5.11), it follows that D c j h j is a bounded sequence in L ([0 , H k − ( D ; R m )). Therefore, in order to prove (5.14), it is enough tocheck it with respect to smooth test functions. Let u ∈ C ([0 , C ∞ ( D ; R m )),and denote w = u + ( − k − ∂ k − θ u ; we then have (cid:10) D c j h j − D c ∗ h, u (cid:11) L ([0 , H k − ( D ; R m )) = (cid:10) D c j h j − D c ∗ h, w (cid:11) L ([0 , L ( D ; R m )) . Since | ∂ θ c j | − → | ∂ θ c ∗ | − uniformly, the right-hand side converges to zero if (cid:10) ∂ θ h j − ∂ θ h, w (cid:11) L ([0 , L ( D ; R m )) → , (cid:10) II c j ( ∂ θ c j , h j ) − II c ∗ ( ∂ θ c ∗ , h ) , w (cid:11) L ([0 , L ( D ; R m )) → . The first one follows from (5.13). The second one follows also from (5.13),using in additon the fact that c j → c ∗ in C ([0 , C n − ( D ; R m )) implies thatII c j → II c ∗ uniformly, and ∂ θ c j → ∂ θ c ∗ uniformly. This completes the proofof (5.14), and hence also of (5.12). Step V: c ∗ is a minimizer. Using the embedding ι , and considering allcurves as curves in R m , we can write the energy as E ( c ) = n X k =0 Z Z π a k ( ℓ c ) | D kc ˙ c | | ∂ θ c | d θ d t = n X k =0 k p a k ( ℓ c ) p | ∂ θ c | D kc ˙ c k L ([0 , L ( D ; R m )) , where the transition to the second line uses the fact that ι is an isometric em-bedding. Since c j → c ∗ in C ([0 , C n − ( D ; R m )), we have that p a k ( ℓ c j ) → OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 27 p a k ( ℓ c ∗ ) uniformly (for k = 0 , . . . , n ), and that p | ∂ θ c j | → p | ∂ θ c ∗ | uni-formly. Therefore, (5.12) implies that for all k = 0 , . . . , n , p a k ( ℓ c j ) q | ∂ θ c j | D kc j ˙ c j ⇀ p a k ( ℓ c ∗ ) p | ∂ θ c ∗ | D kc ∗ ˙ c ∗ in L ([0 , L ( D ; R m )) . Since the map x
7→ k x k in a Hilbert space is weakly sequentially lowersemicontinuous, we obtain thatinf A c ,c E ≤ E ( c ∗ ) ≤ lim inf E ( c j ) → inf A c ,c E, hence c ∗ is a minimizer.5.6. Geodesic completeness in the smooth category.
For closed curves,i.e., D = S we obtain also completeness in the smooth category using theno-loss-no-gain result. Corollary 5.13.
Let n ≥ and let G be a smooth Riemannian metric on I n ( S , N ) . Assume that for every metric ball B ( c , r ) ∈ ( I n ( S , N ) , dist G ) ,there exists a constant C = C ( c , r ) > such that conditions from theorem5.1 hold, i.e., k h k G c ≥ Cℓ − / c k∇ ∂ s h k L ( d s ) , (5.1) k h k G c ≥ C k∇ k∂ s h k L ∞ k = 0 , . . . , n − , (5.2) k h k G c ≥ C k∇ n∂ s h k L . (5.3) Then the space (Imm( S , N ) , G | Imm( S , N ) ) is geodesically complete, where G | Imm( S , N ) is the restriction of the metric G to the space of smooth immer-sions.Proof. The proof of this result follows directly by applying Lemma A.1, for V = T I n ( D, N ), an open subset of H n ( D, T N ), and F the exponential mapof G . (cid:3) For open curves D = [0 , π ] one has to be slightly more careful, due tothe potential loss of smoothness at the boundary; in this case Lemma A.1only yields that solutions to the geodesic equation with smooth initial dataremain at all times in I n ([0 , π ] , N ) ∩ C ∞ loc ((0 , π ) , N ).6. Incompleteness of constant coefficient metrics on opencurves
In our main result we have seen a significant difference between open andclosed curves: while we prove that the constant coefficient metrics of order n ≥ R d it has been observed in [4,Remark 2.7] that constant coefficient Sobolev metrics are in fact metrically-incomplete, by constructing an explicit example of a path that leaves thespace in finite time. Essentially, they showed that one can shrink a straightline to a point using finite energy. This behavior does not appear for closed curves as blow-up of curvature is an obstruction and thus ensures the com-pleteness of the space. The goals of this section are twofold:(1) to extend the example of metric incompleteness from [4] (Exam-ple 6.1);(2) to show that shrinking to a point is the only possibility to leave thespace with finite energy (Theorem 6.3), and deduce from it a con-dition that ensures the existence of geodesics between given curves(Theorem 6.7).The following example of metric incompleteness is a generalization of theexample given in [4, Remark 2.7]. We only present it for R -valued curvesfor the sake of clarity; it can be adapted easily to arbitrary target manifolds(disappearing along a geodesic instead of a straight line). Example 6.1.
Consider I n ([0 , π ]; R ) with the metric k h k G c = k h k L ( d s ) + k∇ n∂ s h k L ( d s ) . Consider the path c : [0 , → I n , defined by c ( t, θ ) = ((1 − t )( θ − π ) + f ( t ) , g ( t ))for some smooth functions f, g : [0 , → R to be determined. Note that c θ = (1 − t, , c t = ( − ( θ − π ) + f ′ ( t ) , g ′ ( t )) , ∇ ∂ s c t = (cid:18) − − t , (cid:19) , ∇ k∂ s c t = 0 for k > . Hence k c t k G c = Z π (cid:0) ( f ′ ( t ) − ( θ − π )) + g ′ ( t ) (cid:1) (1 − t ) d θ = 2 π (1 − t ) (cid:18) π f ′ ( t ) + g ′ ( t ) (cid:19) , and thereforelength( c ) = Z k c t k G c = √ π Z (1 − t ) / (cid:18) π f ′ ( t ) + g ′ ( t ) (cid:19) / d t ≤ √ π Z (1 − t ) / (cid:18) π √ | f ′ ( t ) | + | g ′ ( t ) | (cid:19) d t, hence length( c ) < ∞ if R | f ′ ( t ) | (1 − t ) / d t < ∞ and similarly for g . Underthese restrictions on f and g many things can happen, for example:(1) For f = g = 0 we obtain that c converges, as t →
1, to the constantcurve at the origin;(2) For f ( t ) = tx and g ( t ) = ty , c converges to the constant curve at( x , y ).(3) For f ( t ) = − log(1 − t ) and g = 0, c converges to a point at infinityat the positive end of the x axis. This is true for metrics of order n ≥ n < OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 29 (4) For f ( t ) = sin( − log(1 − t )) and g = 0, c does not converge pointwiseto anything in R .Note that this analysis does not change if we replace G with anotherconstant coefficient metric. This shows that ( I n ([0 , π ]; R ) , dist G ) is notmetrically complete. However, from the point of view of the metric com-pletion, all these different choices of f and g are the same point in thecompletion — indeed, let c i ( t, θ ) = ((1 − t )( θ − π ) + f i ( t ) , g i ( t )) , i = 1 , , and define, for a fixed t ∈ [0 , γ t ( τ, θ ) as the affine homotopybetween c ( t, · ) = γ t (0 , · ) and c ( t, · ) = γ t (1 , · ), that is, γ t ( τ, θ ) = ((1 − t )( θ − π ) + τ f ( t ) + (1 − τ ) f ( t ) , τ g ( t ) + (1 − τ ) g ( t )) . Since | γ tθ | = 1 − t and γ tτ = ( f ( t ) − f ( t ) , g ( t ) − g ( t )) is independent of θ and τ , it follows immediately thatlength( γ t ) ∝ − t. Therefore, dist G ( c ( t, · ) , c ( t, · )) ≤ length( γ t ) ∝ − t → t →
1. This means, that in the metric completion, all the Cauchy se-quences obtained by choosing different f s and g s are equivalent, hence con-verge to a single point.This example leads to the following open question: Question 6.2.
Let G be a constant coefficient Sobolev metric of order n ≥ I n ([0 , π ] , N ). For i = 1 ,
2, let c in ∈ I n ([0 , π ] , N ) betwo Cauchy sequences with ℓ c in →
0. Does it hold thatlim n →∞ dist G ( c n , c n ) = 0?We now show that if a Cauchy sequence of curves does not converge, itslengths must tend to zero: Theorem 6.3.
Let G be a constant coefficient Sobolev metric of order n ≥ of the type (1.2) on I n ([0 , π ]; N ) , where both a and a n are strictly positiveconstants. Assume that ( c n ) n ∈ N ⊂ I n ([0 , π ]; N ) is a Cauchy sequence withrespect to dist G , whose lengths are bounded from below, that is ℓ c n > δ > for all n . Then c n converges to some c ∞ ∈ I n ([0 , π ]; N ) . Before proving this result we note a consequence of it: if the answer toQuestion 6.2 is positive, then, together with Theorem 6.3, it would givea positive answer to the following conjecture on the metric completion of( I n ([0 , π ] , N ) , dist G ): Question 6.4.
Let G be a constant coefficient Sobolev metric of order n ≥ I n ([0 , π ] , N ). Is the metric completion of( I n ([0 , π ] , N ) , dist G ) given by I n ([0 , π ] , N ) ∪ { } , where { } representsthe limit of all vanishing-length Cauchy sequences? In our infinite dimensional situation metric incompleteness does not implygeodesic incompleteness. Furthermore the paths constructed in Example 6.1are not geodesics (a direct calculations shows that the boundary equationsin the geodesic equations are not satisfied). This leads to the followingquestion:
Question 6.5.
Let G be a constant coefficient Sobolev metric of order n ≥ I n ([0 , π ] , N ). Is I n ([0 , π ] , N ) geodesically complete?We proceed with the proof of Theorem 6.3. We will need the followinglemma, which is similar to Lemma 5.9: Lemma 6.6.
Let G be a Sobolev metric of order n ≥ on I n ([0 , π ] , N ) ,such that, for every h ∈ H n ([0 , π ] , c ∗ T N ) , k∇ ∂ s h k L ( d s ) ≤ C max (cid:8) , ℓ − c (cid:9) k h k G c for some uniform constant C > . Then, the function c ℓ / c is Lip-schitz continuous on every metric ball in ( I n ([0 , π ] , N ) , dist G ) . Moreover,the Lipschitz constant of in B ( c , r ) depends only on ℓ c and r , and is an in-creasing function of both, that is, there exists a function L( C, ℓ, r ) , increasingin all variables, such that | ℓ / c − ℓ / c | ≤ L( C, ℓ c , r ) dist G ( c, ˜ c ) for every c, ˜ c ∈ B ( c , r ) . Proof of Lemma 6.6.
As in the proof of Lemma 5.9, we have | D c,h ℓ / c | ≤ ℓ c k∇ ∂ s h k L ( d s ) ≤ C max { ℓ c , } k h k G c ≤ C (1 + ℓ / c ) k h k G c from which the claim follows by Lemma 5.7, with L( C, ℓ, r ) := L ( C, ℓ / , r ). (cid:3) Proof of Theorem 6.3.
Assume that c n is a Cauchy sequence with ℓ c n > δ for some δ > G has constant coefficients (with a , a n > k = 1, that k∇ ∂ s h k L ( d s ) ≤ C max (cid:8) , ℓ − c (cid:9) (cid:16) k h k L ( d s ) + k∇ n∂ s h k L ( d s ) (cid:17) ≤ C ′ max (cid:8) , ℓ − c (cid:9) k h k G c , where the constants C, C ′ depend only on n , a and a n , hence we can applyLemma 6.6.There exists N large enough such that c n ∈ B ( c N , /
2) for all n ≥ N .Applying Lemma 6.6, for B ( c N ,
1) we obtain that there exists a constant ¯ ℓ ,depending on c N such that ℓ c ≤ ¯ ℓ for all c ∈ B ( c N , . In particular, this applies to all c n for n ≥ N .Let L( C ′ , ℓ, r ) be the Lipschitz constant bound as in Lemma 6.6, anddenote ¯ L := L( C ′ , ¯ ℓ, r := min n δ / L , / o . There exists anindex N > N such that for n ≥ N we have that c n ∈ B ( c N , r / OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 31 is dist G ( c n , c N ) < r /
3. Applying Lemma 6.6 to B ( c N , r ) and the bound ℓ c N ≤ ¯ ℓ , we have that (cid:12)(cid:12)(cid:12) ℓ / c − ℓ / c (cid:12)(cid:12)(cid:12) ≤ ¯ L dist G ( c, ˜ c ) for every c, ˜ c ∈ B ( c N , r ) . Since ℓ c N > δ , and B ( c N , r ) ⊂ B ( c N , ℓ c ∈ (cid:20) δ / , ¯ ℓ (cid:21) for every c ∈ B ( c N , r ) . Denote by G ′ the standard scale-invariant metric of order n on I n ([0 , π ] , N );that is, k h k G ′ c = ℓ − c k h k L ( d s ) + ℓ n − c k∇ n∂ s h k L ( d s ) . Recall that ( I n , dist G ′ ) is metrically complete by Theorem 5.3. Using (6.1)and Lemma 4.1, it follows that G ′ and G are equivalent in B ( c N , r ). Fromhere we continue in a similar way as in Propositions 5.5–5.6: Let c, ˜ c ∈ B ( c N , r / < ε < r / − dist G ( c, ˜ c ). Let γ be a curve between c and ˜ c such that length G ( γ ) < dist G ( c, ˜ c ) + ε . By triangle inequality, we havethat γ ⊂ B ( c N , r ), and since G ′ and G are equivalent there, we have thatfor some constant C > γ ),dist G ′ ( c, ˜ c ) ≤ length G ′ ( γ ) ≤ C length G ( γ ) < C (dist G ( c, ˜ c ) + ε ) , and since ε is arbitrarily small, we conclude thatdist G ′ ( c, ˜ c ) ≤ C dist G ( c, ˜ c ) , for every c, ˜ c ∈ B ( c N , r / . Since for every n ≥ N , c n ∈ B ( c N , r / c n is a Cauchysequence with respect to G ′ as well. Since ( I n , dist G ′ ) is metrically complete,we have that there exists c ∞ ∈ I n such that dist G ′ ( c n , c ∞ ) →
0. Since both G and G ′ are strong metrics on I n , they induce the same topology [24, VII,Proposition 6.1], and thus c n → c ∞ ∈ I n with respect to G as well, whichcompletes the proof. (cid:3) From the arguments in the proof of Theorem 6.3, we also obtain that forclose enough immersions c , c ∈ I n ([0 , π ]; N ), there exists a connectingminimizing geodesic: Theorem 6.7.
Let G be a constant coefficient Sobolev metric of order n ≥ of the type (1.2) on I n ([0 , π ]; N ) , where both a and a n are strictly positiveconstants. Let c ∈ I n ([0 , π ]; N ) . Then, there exists a constant r , depend-ing only on the coefficients a k and on ℓ c , such that for every c ∈ B ( c , r ) ,there exists a minimizing geodesic between c and c . Remark 6.8.
The proof below, together with the bound (5.5), imply that r can be chosen such that r = C ℓ / c ℓ / c ≥ C min (cid:18) ℓ / c , (cid:19) , where C depends only on the coefficients a k , k = 0 , . . . , n . Note that wedo not know whether the existence of minimizing geodesics fails in general;it might be that although the space in metrically incomplete, a minimizinggeodesic between any two curves c , c ∈ I n ([0 , π ] , N ) exists. Proof.
As in Theorem 6.3, there exists a constant C , depending only on n , a and a n (or alternatively, on a k , k = 0 , . . . , n ) such that the assumptionof Lemma 6.6 holds. Fix ˜ L := L( C, ℓ c , C, ℓ, r ) is the Lipschitzconstant function from Lemma 6.6. Let r = min ℓ / c L , ! . It follows that ℓ c ∈ " / ℓ c , / / ℓ c for every c ∈ B ( c , r ) . As in Theorem 6.3, it follows that in this ball G is uniformly equivalent toa scale-invariant Sobolev metric of order n on I n ([0 , π ]; N ), hence Lem-mata 5.10–5.12 holds uniformly on B ( c , r ) (rather than on every metricball).Let c ∈ B ( c , r ). Define the energy E ( c ) and the set of paths A c ,c asin Section 5.5. Let c ∈ A c ,c , with length( c ) < r . Assume that c is hasconstant speed; we then have E ( c ) = length( c ) < r . Therefore, inf A c ,c E < r . We can now take a minimizing sequence c j ∈ A c ,c , and assume withoutloss of generality that E ( c j ) < r for all j . The proof now follows in thesame way as in Theorem 6.3. (cid:3) Appendix A. The geodesic equation
A.1.
Proof of Lemma 3.3: the geodesic equation.
Proof of Lemma 3.3.
To prove the formula for the geodesic equation we con-sider of the energy of a path of immersions c ( t, θ ). Furthermore, we will treatthe zero and first order terms separately. Varying c ( t, θ ) in direction h ( t, θ )with h (0 , θ ) = h (1 , θ ) = 0 we obtain for the zeroth order term: d (cid:18)Z a ( ℓ c ) Z D g ( c t , c t ) | c ′ | d θ d t (cid:19) ( h )= Z a ′ ( ℓ c ) D c,h ℓ c Z D g ( c t , c t ) | c ′ | d θ d t + Z a ( ℓ c ) Z D g ( ∇ h c t , c t ) + g ( c t , c t ) g ( v, ∇ s h ) d s d t = Z a ′ ( ℓ c ) Z D g ( v, ∇ ∂ s h ) d s Z D g ( c t , c t ) d s d t + Z a ( ℓ c ) Z D g ( ∇ ∂ t h, c t ) + g ( ∇ s h, vg ( c t , c t )) d s d t OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 33 where we used in the last step that(A.1) ∇ h c t = ∇ ∂ t h . and the variation formula for the length ℓ c from Lemma 3.2.Similarly we calculate for the first order terms: d (cid:18)Z a ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) | c ′ | d θ d t (cid:19) ( h )= Z a ′ ( ℓ c ) Z D g ( v, ∇ ∂ s h ) d s Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s d t + Z a ( ℓ c ) Z D g ( ∇ h ∇ ∂ s c t , ∇ ∂ s c t ) + g ( ∇ ∂ s c t , ∇ ∂ s c t ) g ( v, ∇ s h ) d s d t = Z a ′ ( ℓ c ) Z D g ( v, ∇ ∂ s h ) d s Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s d t + Z a ( ℓ c ) Z D g ( − g ( v, ∇ ∂ s h ) ∇ ∂ s c t + ∇ ∂ s ∇ h c t + R ( v, h ) c t , ∇ ∂ s c t ) d s d t + Z a ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) g ( v, ∇ s h ) d s d t = Z a ′ ( ℓ c ) Z D g ( v, ∇ ∂ s h ) d s Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s d t + Z a ( ℓ c ) Z D g ( − g ( v, ∇ ∂ s h ) ∇ ∂ s c t + 2 ∇ ∂ s ∇ ∂ t h + 2 R ( v, h ) c t , ∇ ∂ s c t ) d s d t Sorting this by derivatives of h we obtain d (cid:18)Z a ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) | c ′ | d θ d t (cid:19) ( h )= Z a ( ℓ c ) Z D g ( ∇ ∂ s ∇ ∂ t h, ∇ ∂ s c t ) + 2 g ( R ( v, h ) c t , ∇ ∂ s c t )+ g (cid:18) ∇ ∂ s h, a ′ ( ℓ c ) a ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s v − g ( ∇ ∂ s c t , ∇ ∂ s c t ) v (cid:19) d s d t Putting both together we obtain: dE ( c ) .h = Z Z D a ( ℓ c ) g ( ∇ ∂ t h, c t ) + 2 a ( ℓ c ) g ( ∇ ∂ s ∇ ∂ t h, ∇ ∂ s c t )+ 2 a ( ℓ c ) g ( R ( v, h ) c t , ∇ ∂ s c t ) + g ( ∇ ∂ s h, Ψ c ( c t , c t ) v ) d s d t whereΨ c ( c t , c t ) = a ( ℓ c ) g ( c t , c t ) + a ′ ( ℓ c ) Z D g ( c t , c t ) d s − a ( ℓ c ) g ( ∇ ∂ s c t , ∇ ∂ s c t ) + a ′ ( ℓ c ) Z D g ( ∇ ∂ s c t , ∇ ∂ s c t ) d s. To obtain the geodesic equation, we have to integrate by parts to free h from all derivatives. We will treat the four terms separately. For the firsttwo terms we recall that d s depends on the curve c (and thus on time t ), i.e., for time dependent vector fields h and k , with h (0 , θ ) = h (1 , θ ) = 0, wehave Z Z D g ( ∇ ∂ t h, k ) d s d t = − Z Z D g ( h, ∇ ∂ t ( | c ′ | k )) d θ d t = − Z Z D g ( h, ∇ ∂ t k + g ( v, ∇ ∂ s c t ) k ) d s d t . (A.2)Applying this formula to the first term yields:2 Z Z D g ( ∇ ∂ t h, a ( ℓ c ) c t ) d s d t = − Z D a ( ℓ c ) g (cid:16) h, ∇ t c t + g ( v, ∇ ∂ s c t ) c t + a ′ ( ℓ c ) a ( ℓ c ) Z D g ( v, ∇ ∂ s c t ) d s c t (cid:17) d s d t . For the second term we need to apply integration by parts in space first:2 Z Z D g ( ∇ ∂ s ∇ ∂ t h, a ( ℓ c ) ∇ ∂ s c t ) d s d t = 2 Z g ( ∇ ∂ t h, a ( ℓ c ) ∇ ∂ s c t ) | π d t − Z Z D g ( ∇ ∂ t h, a ( ℓ c ) ∇ ∂ s ∇ ∂ s c t ) d s d t = − Z g ( h, ∇ ∂ t ( a ( ℓ c ) ∇ ∂ s c t )) | π d t + 2 Z Z D a ( ℓ c ) g ( h, ∇ ∂ t ∇ s c t ) d s d t + 2 Z Z D a ( ℓ c ) g (cid:18) h, g ( v, ∇ ∂ s c t ) ∇ s c t + a ′ ( ℓ c ) a ( ℓ c ) Z D g ( v, ∇ ∂ s c t ) d s ∇ s c t (cid:19) d s d t For the third term we use the symmetries of the curvature tensor to obtain2 Z Z D a ( ℓ c ) g ( R ( v, h ) c t , ∇ ∂ s c t ) d s d t = 2 Z Z D a ( ℓ c ) g ( R ( c t , ∇ ∂ s c t ) v, h ) d s d t. Finally for the last term we need to integrate in parts in space again, takinginto account the boundary terms: Z Z D g ( ∇ ∂ s h, Ψ c ( c t , c t ) v ) d s d t = Z g (cid:0) h, Ψ c ( c t , c t ) v ) (cid:12)(cid:12)(cid:12) π d t − Z Z D g (cid:0) h, ∇ ∂ s (Ψ c ( c t , c t ) v ) (cid:1) d s d t . We can now read off the geodesic equation. We will fist start by collectingthe terms on the interior of D : a ( ℓ c ) ∇ ∂ t c t − a ( ℓ c ) ∇ ∂ t ∇ s c t = − a ( ℓ c ) g ( v, ∇ ∂ s c t ) c t − a ′ ( ℓ c ) (cid:18)Z D g ( v, ∇ ∂ s c t ) d s (cid:19) c t + a ( ℓ c ) g ( v, ∇ ∂ s c t ) ∇ ∂ s c t + a ′ ( ℓ c ) (cid:18)Z D g ( v, ∇ ∂ s c t ) d s (cid:19) ∇ s c t + a ( ℓ c ) R ( c t , ∇ ∂ s c t ) v − ∇ ∂ s (Ψ c ( c t , c t ) v ) . OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 35
From here the result follows using the definition of the inertia operator A c ,the product rule for the term ∇ ∂ s (Ψ c ( c t , c t ) v ), by using the formula ∇ t ( A c c t ) = ( ∇ t A c ) c t + A c ( ∇ t c t ) = ∇ t ( a ( ℓ c ) c t − a ( ℓ c ) ∇ ∂ s c t )= (cid:18)Z D g ( v, ∇ ∂ s c t ) d s (cid:19) a ′ ( ℓ c ) c t + a ( ℓ c ) ∇ ∂ t c t − (cid:18)Z D g ( v, ∇ ∂ s c t ) d s (cid:19) a ′ ( ℓ c ) ∇ s c t − a ( ℓ c ) ∇ ∂ t ∇ s c t , and by collecting the boundary terms if D = [0 , π ]. (cid:3) A.2.
Proof of Theorem 3.8: local well-posedness.
In this section wewill use the method of Ebin-Marsden to obtain local well-posedness anduniqueness of the geodesic equations. Before we will prove the local well-posedness we will formulate a variant of the no-loss-no-gain result, which isalso used in Section 5.6.
Lemma A.1.
Let q ≥ , V ⊂ H q ( D, T N ) an open subset and let F : V → H q ( D, T N ) be a smooth and D q ( D ) equivariant map, i.e., F ( h ◦ ϕ ) = F ( h ) ◦ ϕ for all h ∈ H q ( D, T N ) and ϕ ∈ D q ( D ) . Then F is a smooth map from V ∩ H q + l loc ( D o , T N ) to itself for any l ∈ N , where D o is the interior of D .Proof. For D = S this result is shown in [13, Corollary 4.1]. For the case D = [0 , π ] the proof is essentially the same, see also the arguments ofEbin and Marsden [20, Theorem 12.1, Lemma 12.2] who proved the originalno-loss-no-gain results for manifolds with boundary. (cid:3) Proof of Theorem 3.8.
For closed curves, i.e., D = S , this result can befound in [10, Theorem 4.4], see also [28, 6]. In the following we will focuson the case of open curves, where the proof will be slightly more involveddue to the existence of a boundary. For a strong Riemannian metric ( q = n )the existence of the geodesic equation and its local well-posedness is alwaysguaranteed, see, e.g., [24, VIII, Theorem 4.2]. Thus we obtain the first partof the theorem for the Sobolev metric of order n ≥ I n ([0 , π ] , N ) byTheorem 3.7. For q = n we have to prove the well-posedness by hand. In thefollwoing we will assume that n = 1; the proof for n > A c under Neumann boundary conditions: Claim: Let f ∈ H r I q ([0 , π ] , T N ) and c ∈ I q ([0 , π ] , N ) with q − ≥ r ≥ and π ◦ f = c . Then the boundary value problem (A.3) A c u ( θ ) = f ( θ ) , ∇ ∂ θ u (0) = u , ∇ ∂ θ u (2 π ) = u has a unique solution u ∈ H r +2 I q ([0 , π ] , T N ) , with π ◦ u = c . Note that by subtracting any H q section that satisfy the boundary condi-tions, we can assume that the boundary conditions are homogeneous. Then,a weak form of this equation is simply G c ( u, w ) = R π g ( f, w ) d θ for every w ∈ H ([0 , π ] , c ∗ T N ). Since c is fixed, G c is equivalent to the standard H
16 MARTIN BAUER, CY MAOR, AND PETER W. MICHOR norm on H ([0 , π ] , c ∗ T N ). By the Lax-Milgram theorem, there exists aunique solution u ∗ ∈ H . We can then consider the equation A c u ( θ ) = f ( θ )with initial conditions u (0) = u ∗ (0), ∇ ∂ θ u (0) = 0. By moving to the weakform again, it follows that the solution for this initial value problem mustbe u ∗ , and so its regularity then follows from standard initial-value ODEtheory. This completes the proof of this claim.To apply this theorem to the geodesic equation we need to observe thatfor any fixed time t the boundary terms of the geodesic equation can berewritten to yield exactly Neumann conditions for the system A c ( ∇ ∂ t c t ) = (cid:18) − ( ∇ ∂ t A c ) c t − g ( v, ∇ ∂ s c t ) A c c t −
12 Ψ c ( c t , c t ) ∇ ∂ s v − g ( ∇ ∂ s c t , A c c t ) v + a ( ℓ c ) R ( c t , ∇ ∂ s c t ) v (cid:19) , where ( ∇ ∂ t A c ) = ∇ ∂ t ◦ A c − A c ◦ ∇ ∂ t , which is an operator of order 2. Inaddition we have the boundary conditions ∇ θ ∇ ∂ t c t (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = F ( c, c t ) ∈ R ∇ θ ∇ ∂ t c t (cid:12)(cid:12)(cid:12)(cid:12) θ =2 π = F ( c, c t ) ∈ R . where F and F can be calculated by applying the product formula fordifferentiation and the formula for swapping covariant derivatives to theboundary conditions in Lemma 3.3.Thus by the claim above we can invert A c to rewrite the geodesic equationas ∇ ∂ t c t = A − c (cid:18) − ( ∇ ∂ t A c ) c t − g ( v, ∇ ∂ s c t ) A c c t −
12 Ψ c ( c t , c t ) ∇ ∂ s v − g ( ∇ ∂ s c t , A c c t ) v + a ( ℓ c ) R ( c t , ∇ ∂ s c t ) v (cid:19) . The right hand side of this equation defines a smooth mappingΦ : T I q ( D, N ) → T I q ( D, N ) , where the smoothness of Φ follows directly by counting derivatives, using theSobolev embedding theorem and the result that A c and thus also ( ∇ ∂ t A c )and A − c are smooth. Thus we have interpreted the geodesic equation as anODE (in t ) on a Banach space of functions. From here the proof of item 1of Theorem 3.8 follows directly as in [10, Theorem 4.4] and reduces to anapplication of the Cauchy theorem and the equivalence of fiber-wise qua-dratic smooth mappings Φ : T I q ( D, N ) → T I q ( D, N ) and smooth sprays S : T I q ( D, N ) → T T I q ( D, N ).To prove item 2 of Theorem 3.8, we use Lemma A.1, for F the exponentialmap G on I q ( D, N ), and V ⊂ H q I q ( D, T N ) is a neighborhood of the zerosection on which the exponential map is defined. It follows that the domainof existence of the geodesic equation (in t ) and the neighborhoods for theexponential mapping are uniform in the Sobolev exponential l ∈ N and thus OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 37 the result continues to hold on I q + l loc ( D, N ) and therefore also locally in thesmooth category. (cid:3) Appendix B. Holonomy estimates: proof of Lemma 4.2
We now prove the Sobolev estimates for manifolds-valued curves as statedin Lemma 4.2. We start be proving some geometric estimates, culminatingin bounds on the holonomy along a closed curve (Proposition B.3). Thesettings for the geometric estimates is as follows:Let ( N , g ) be a complete Riemannian manifold of finite dimension, withbounded sectional curvature, | K | ≤ K N and positive injectivity radiusinj N >
0. We denote by R the Riemann curvature of g .Let c : [0 , a ] → N be a curve, and let V be a vector field along c . LetΠ θ θ : T c ( θ ) N → T c ( θ ) N be the parallel transport operator along c , and Ddθ the covariant derivative along c . Lemma B.1. | V ( a ) − Π a V (0) | ≤ Z a (cid:12)(cid:12)(cid:12)(cid:12) Ddθ V ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) d θ. Proof.
Define f ( θ ) = Π aθ V ( θ ) − Π a V (0). Our goal is to bound | f ( a ) | . Notethat f (0) = 0, and that ∂∂θ f ( θ ) = Π aθ Ddθ V ( θ ) . Therefore, using the fact that the parallel transport is an isometry, we have | f ( a ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z a ∂∂θ f ( θ ) d θ (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z a (cid:12)(cid:12)(cid:12)(cid:12) Ddθ V ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) d θ. (cid:3) Let c : [0 , a ] → N be a closed curve, c (0) = c ( a ) = p , with ℓ c < N .Define a map c ( θ, t ) : [0 , a ] × [0 , → N , such that c ( θ, · ) is the uniquegeodesic connecting p and c ( θ ). This is well defined since ℓ c < N impliesthat dist( p, c ( θ )) < ℓ c / < inj N for any θ . In other words, if we define γ ( θ ) = exp − p ( c ( θ )), then c ( θ, t ) := exp p ( tγ ( θ )). For every t ∈ [0 , c t := c ( · , t ) : [0 , a ] → N is a closed curve based in p , and for t = 0 it is theconstant curve. Lemma B.2.
There exists a constant C , depending only on inj N and theupper bound for the sectional curvature of N , such that for curve c with ℓ c < C , then ℓ c t ≤ ℓ c for every t ∈ [0 , .Proof. In the following we will assume that ℓ c < N , otherwise the family c t is not well-defined.It is obviously sufficient to prove that | ∂ θ c ( θ, t ) | ≤ | ∂ θ c ( θ, | for every θ and t . Note that for a fixed θ , J ( t ) := ∂ θ c ( θ , t ) is a Jacobi field, hence itsatisfies the Jacobi equation D dt J + R ( J, ∂ t c ( θ , t )) ∂ t c ( θ , t ) = 0 , with the initial conditions J (0) = 0 , Ddt J (0) = ∂∂θ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ exp − p c ( θ ) =: γ ′ ( θ ) . These initial conditions follow from the fact that J (0) = ∂∂θ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ c ( θ,
0) = ∂∂θ (cid:12)(cid:12)(cid:12)(cid:12) θ = θ p = 0 , and Ddt J (0) =
D∂t ∂∂θ c (cid:12)(cid:12)(cid:12)(cid:12) ( θ,t )=( θ , = D∂θ ∂∂t c (cid:12)(cid:12)(cid:12)(cid:12) ( θ,t )=( θ , = D∂θ d tγ ( θ ) exp p [ γ ( θ )] (cid:12)(cid:12)(cid:12)(cid:12) ( θ,t )=( θ , = D∂θ d exp p [ γ ( θ )] (cid:12)(cid:12)(cid:12)(cid:12) θ = θ = D∂θ γ ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) θ = θ = γ ′ ( θ ) , where we used the fact that d exp p = id T p N , and that when t = 0, c ( θ,
0) = p for all θ , hence covariant derivative along θ is the same as the regularderivative in id T p N .Note that we can always reparametrize θ such that | γ ′ ( θ ) | = 1 for any θ ,hence (cid:12)(cid:12) Ddt J (0) (cid:12)(cid:12) = 1.Our aim is to prove that | J ( t ) | ≤ | J (1) | . The proof mimics the proof ofRauch’s comparison theorem. Define f ( t ) := | J ( t ) | ; we want to prove that˙ f ( t ) ≥ t ∈ (0 , J := Ddt J , ¨ J := D dt J . We then have˙ f = g ( J, ˙ J ) | J | . We have J (0) = 0 and therefore, by the Jacobi equations, also ¨ J (0) = 0.We therefore obtain that˙ J ( t ) = ˙ J (0) + O ( t ) , J ( t ) = t ˙ J (0) + O ( t ) , hence ˙ f ( t ) = 1 t + O ( t ) . Using the Jacobi equations and the upper bound K on the sectional curva-ture of N , we obtain¨ f ( t ) = (cid:16) | ˙ J | + g ( J, ¨ J ) (cid:17) | J | − g ( J, ˙ J ) | J | ≥ g ( J, ¨ J ) | J | − g ( J, ˙ J ) | J | = g ( J, ¨ J ) | J | − ˙ f = − g ( J, R ( J, ∂ t c ) ∂ t c ) | J | − ˙ f ≥ − K | ∂ t c | | J | ) | J | − ˙ f = − K | ∂ t c | − ˙ f ≥ − K dist N ( p, c ( θ )) − ˙ f ≥ − Kℓ c − ˙ f , where we used the fact that | ∂ t c ( θ , t ) | = dist N ( p, c ( θ )) since c ( θ , t ) is aconstant speed geodesic from p to c ( θ ). We obtain that¨ f ( t ) + ˙ f ≥ − Kℓ c , ˙ f ( t ) = 1 t + O ( t ) . OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 39
From the Riccati comparison estimate [31, Corollary 6.4.2], it follows thatfor t > f ( t ) ≥ √ Kℓ c cot (cid:16) √ Kℓ c t (cid:17) K > , t ≤ π √ Kℓ c t K = 0 √− Kℓ c coth (cid:16) √− Kℓ c t (cid:17) K < . If K ≤
0, it follows that ˙ f ( t ) > t >
0, and we are done. If
K > ℓ c < π/ √ K , we obtain that ˙ f ( t ) is larger than a functionthat is positive in (0 , (cid:3) We now state the main geometric estimate we need. Recall that, in twodimensions, the holonomy of a small closed curve is roughly the area ofenclosed by the curve times the curvature inside it, and that by the isoperi-metric inequality, the area grows at most like the length of the curve squared.The following proposition combines these statements (in any dimension) intoa quantitative estimate on the holonomy:
Proposition B.3.
There exists a constant C = C ( K N , inj N , dim N ) > ,such that for every closed curve c ⊂ N based in T p N , (cid:12)(cid:12) Hol c − id T p N (cid:12)(cid:12) ≤ min n Cℓ c , √ dim N o where Hol c is the holonomy along c and ℓ c is the length of the curve c .Proof. Since Hol c is an isometry of T p N , | Hol c | = | id T p N | = √ dim N .Therefore, by triangle inequality, we have (cid:12)(cid:12) Hol c − id T p N (cid:12)(cid:12) ≤ √ dim N .In the following, we assume that C ≥ √ dim N /C , where C is defined inLemma B.2, and therefore it is sufficient to prove that (cid:12)(cid:12) Hol c − id T p N (cid:12)(cid:12) ≤ Cℓ c under the assumption that ℓ c ≤ C .Fix v ∈ T p N a unit vector. Our goal is to prove that | Hol c v − v | ≤ Cℓ c . Define the family of curves c ( θ, t ) = c t ( θ ) : [0 , a ] × [0 , → N as inLemma B.2. Define a vector field X ∈ Γ( c ∗ T N ) by X ( θ, t ) := Π c t ( θ ) p v, where Π c t ( θ ) p : T p N → T c t ( θ ) N is the parallel transport along the curve c t .We have X ( θ,
0) = v, X (0 , t ) = v, X ( a, t ) = Hol c t v. Since c ( a, t ) = p , the parallel transport along the curve c ( a, · ) is the identity,and so, by Lemma B.1 we have that | Hol c v − v | = | X ( a, − X ( a, | ≤ Z (cid:12)(cid:12)(cid:12)(cid:12) D∂t X ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) d t. Since c (0 , t ) = p for all t , the covariant derivative D∂t along (0 , t ) is simplythe standard derivative ∂∂t . Therefore, since X (0 , t ) = v does not depend on t , we have D∂t X (0 , t ) = 0. Hence, using Lemma B.1 again, we have (cid:12)(cid:12)(cid:12)(cid:12) D∂t X ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z a (cid:12)(cid:12)(cid:12)(cid:12) D∂θ D∂t X ( θ, t ) (cid:12)(cid:12)(cid:12)(cid:12) d θ. Since X ( θ, t ) is the parallel transport of X (0 , t ) = v along the constant t curve, we have D∂θ X ( θ, t ) = 0, and therefore D∂t D∂θ X ( θ, t ) = 0. Combiningthis with D∂θ D∂t X − D∂t D∂θ X = R (cid:18) ∂∂θ ∂∂t (cid:19) X (see, e.g., [17, Chapter 4, Lemma 4.1]), we have (cid:12)(cid:12)(cid:12)(cid:12) D∂θ D∂t X (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) R (cid:18) ∂c∂θ ∂c∂t (cid:19) X (cid:12)(cid:12)(cid:12)(cid:12) ≤ K N (cid:12)(cid:12)(cid:12)(cid:12) ∂∂θ (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) , where we used the fact that | X | = | v | = 1 since the parallel transport is anisometry. Since c ( θ, · ) is a constant speed geodesic from p to c ( θ ) = c ( θ, p, c ( θ )) ≤ ℓ c /
2, we have that (cid:12)(cid:12)(cid:12)(cid:12) ∂c∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ ℓ c / . Combining these estimates, we obtain | Hol c v − v | ≤ Z (cid:12)(cid:12)(cid:12)(cid:12) D∂t X ( a, t ) (cid:12)(cid:12)(cid:12)(cid:12) d t ≤ Z Z a (cid:12)(cid:12)(cid:12)(cid:12) D∂θ D∂t X (cid:12)(cid:12)(cid:12)(cid:12) d θ d t ≤ K N ℓ c Z Z a (cid:12)(cid:12)(cid:12)(cid:12) ∂c∂θ (cid:12)(cid:12)(cid:12)(cid:12) d θ d t = K N ℓ c Z ℓ c t d t ≤ K N ℓ c , where in the last inequality we used Lemma B.2 to estimate ℓ c t . (cid:3) Using these holonomy estimates, we can now prove Lemma 4.2:
Proof of Lemma 4.2.
As mentioned at the beginning of Section 4, al-though h (0) = h (2 π ) when D = S , it is not true that H (0) = H (2 π ),where H ( θ ) = Π θ h ( θ ) , because of holonomy effects. Therefore, in order to prove (4.4) we cannotuse Sobolev inequalities for periodic functions verbatim, but rather use theresult of Proposition B.3, which implies that for short curves H is ”almost”periodic since the holonomy is small. We will do so by induction over k and n . Base step: the case k = 1 , n = 2 . Assume that k = 1 and n = 2.When ℓ c ≥
1, the inequality (4.2) implies (4.4) by taking a = 1. We are leftto treat the case ℓ c < θ θ the parallel transport from T c ( θ ) N to T c ( θ ) N along c (in the direction dictated by the parameter θ ). Now, byapplying (4.1) for ∇ ∂ s h and using the fundamental theorem of calculus, wehave:Π θ ∇ ∂ s h ( θ ) − ∇ ∂ s h (0) = Z θ ddσ Π σ ∇ ∂ s h ( σ ) d σ = Z θ Π σ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ. Integrating over θ with respect to d s , we obtain ∇ ∂ s h (0) − ℓ c Z S Π θ ∇ ∂ s h ( θ ) d s ( θ ) = − ℓ c Z S Z θ Π σ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ d s ( θ ) . Using again (4.1), we have
OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 41 Z S Π θ ∇ ∂ s h ( θ ) d s ( θ ) = Z π Π θ ∇ ∂ θ h ( θ ) d θ = Z π ddθ Π θ h ( θ ) d θ = Π π h (0) − h (0) , which is not necessarily zero since there the holonomy along c might benon-trivial. We therefore obtain(B.1) ∇ ∂ s h (0) − ℓ c (cid:0) Π π h (0) − h (0) (cid:1) = − ℓ c Z S Z θ Π σ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ d s ( θ ) . Similarly, ∇ ∂ s h (2 π ) − Π πθ ∇ ∂ s h ( θ ) = Z πθ ddσ Π πσ ∇ ∂ s h ( σ ) d σ = Z πθ Π πσ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ, and Z S Π πθ ∇ ∂ s h ( θ ) d s ( θ ) = Z π ddθ Π πθ h ( θ ) d θ = h (2 π ) − Π π h (2 π ) . Thus, using the fact that h (0) = h (2 π ) and ∇ ∂ s h (0) = ∇ ∂ s h (2 π ), we have(B.2) ∇ ∂ s h (0) − ℓ c (cid:0) h (0) − Π π h (0) (cid:1) = 1 ℓ c Z S Z πθ Π πσ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ d s ( θ ) . Adding (B.1) and (B.2), we obtain ∇ ∂ s h (0) − ℓ c (cid:0) Π π h (0) − Π π h (0) (cid:1) = 12 ℓ c Z S (cid:18)Z πθ Π πσ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ − Z θ Π σ ( ∇ ∂ θ ∇ ∂ s h ( σ )) d σ (cid:19) d s ( θ ) . Therefore, using the fact that Π θ θ is an isometry, we obtain that |∇ ∂ s h (0) | ≤ (cid:12)(cid:12) Π π − Π π (cid:12)(cid:12) ℓ c | h (0) | + 12 ℓ c Z S Z π |∇ ∂ θ ∇ ∂ s h ( σ ) | d σ d s ( θ )= (cid:12)(cid:12) Π π − Π π (cid:12)(cid:12) ℓ c | h (0) | + 12 Z S (cid:12)(cid:12) ∇ ∂ s h ( σ ) (cid:12)(cid:12) d s ( σ ) . Using the estimate on the magnitude of the holonomy in Proposition B.3,we have |∇ ∂ s h (0) | ≤ min ( Cℓ c , √ dim N ℓ c ) | h (0) | + 12 Z S (cid:12)(cid:12) ∇ ∂ s h ( σ ) (cid:12)(cid:12) d s, for some C > N . In this inequality the point 0 isarbitrary, hence the above holds for h ( θ ) , ∇ ∂ s h ( θ ) instead of h (0) , ∇ ∂ s h (0). Squaring this inequality, and using the inequality ( a + b ) ≤ a + b ) andCauchy-Schwartz (or Jensen’s) inequality, we obtain, for every θ , |∇ ∂ s h ( θ ) | ≤ min (cid:26) C ℓ c , N ℓ c (cid:27) | h ( θ ) | + ℓ c k∇ ∂ s h k L ( d s ) ≤ C ′ min (cid:26) ℓ c , ℓ c (cid:27) | h ( θ ) | + ℓ c k∇ ∂ s h k L ( d s ) . (B.3)Since we assumed ℓ c ≤
1, Inequality (B.3) implies (4.4) by integrating withrespect to d s . Induction step.
Now assume we have (4.4) for n = 2 , . . . , m and k =1 , . . . , n −
1; we will now prove it for n = m + 1, k = 1 , . . . , m . Denote theconstant in (4.4) by C k,n . Besides k and n , C k,n will depend also on theproperties of the manifold N as stated in the formulation of the Lemma,but we omit this dependence as it is fixed throughout the induction.First, assume k = 1. If ℓ c ≥ min n , (2 C m − ,m C ,m ) − / o , then (4.2) im-plies (4.4) for k = 1 , n = m + 1, by letting a = min n , (2 C m − ,m C ,m ) − / o .If ℓ c ≤ min n , (2 C m − ,m C ,m ) − / o , we have k∇ ∂ s h k L ( d s ) ≤ C ,m ℓ c (cid:16) k h k L ( d s ) + k∇ m∂ s h k L ( d s ) (cid:17) ≤ C ,m ℓ c (cid:16) k h k L ( d s ) + C m − ,m (cid:16) k∇ ∂ s h k L ( d s ) + k∇ m +1 ∂ s h k L ( d s ) (cid:17)(cid:17) , where in the second line we applied the induction hypothesis to ∇ ∂ s h .Moving the C ,m C m − ,m ℓ c k∇ ∂ s h k L ( d s ) to the other side, and noting that C ,m C m − ,m ℓ c ≤ / k∇ ∂ s h k L ( d s ) ≤ C ,m ℓ c (cid:16) k h k L ( d s ) + C m − ,m k∇ m +1 ∂ s h k L ( d s ) (cid:17) , which completes the proof for k = 1.We now assume k >
1. If ℓ c ≥ min n , (2 C k − ,m C ,k ) − / o , then (4.2) im-plies (4.4) for k = 1 , n = m + 1, by letting a = min n , (2 C k − ,m C ,k ) − / o .If ℓ c ≤ min n , (2 C k − ,m C ,k ) − / o , we have (by applying the inductionhypothesis for ∇ ∂ s h ), k∇ k∂ s h k L ( d s ) ≤ C k − ,m ℓ c (cid:16) k∇ ∂ s h k L ( d s ) + k∇ m +1 ∂ s h k L ( d s ) (cid:17) ≤ C k − ,m ℓ c (cid:16) C ,k (cid:16) k h k L ( d s ) + k∇ k∂ s h k L ( d s ) (cid:17) + k∇ m +1 ∂ s h k L ( d s ) (cid:17) . Moving the C k − ,m C ,k ℓ c k∇ k∂ s h k L ( d s ) to the other side, and noting that C k − ,m C ,k ℓ c ≤ / k∇ k∂ s h k L ( d s ) ≤ C k − ,m ℓ c (cid:16) C ,k k h k L ( d s ) + k∇ m +1 ∂ s h k L ( d s ) (cid:17) , which completes the proof for k > OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 43
Appendix C. Proof of Lemma 5.12
First, we note that for a function f ∈ L ( D ) we have, for every c ∈I n ( D, N ), k f k L ( d θ ) ≤ | D | / k f k L ∞ hence boundedness on metric balls of k∇ k∂ s | c ′ |k L ∞ implies boundedness of k∇ k∂ s | c ′ |k L ( d θ ) . Lemma 5.9 implies that under the assumption (5.1), the L ( d θ ) and L ( d s ) norms are equivalent on metric balls, hence boundednesson metric balls of k∇ k∂ s | c ′ |k L ∞ also implies boundedness of k∇ k∂ s | c ′ |k L ( d s ) .Therefore, by Lemma 5.7, our goal is to show that k D c,h ( ∇ k∂ s | c ′ | ) k L p ≤ C (1 + k∇ k∂ s | c ′ |k L p ) k h k G c , where p = ∞ for k = 0 , . . . , n − p = 2 for k = n −
1. We will firstprove the case p = ∞ by induction on k , and then treat the case p = 2, k = n − L ( d θ ) and L ( d s ) are similar, so for brevity,we simply write L ).The claim for k = 0 was proven in Lemma 5.10. We now assume theclaim is true up to k − k . First, note that D c,h ( ∇ k∂ s | c ′ | ) = ∇ k∂ s ( g ( v, ∇ ∂ s h ) | c ′ | ) − k − X i =0 (cid:18) ki + 1 (cid:19) ∇ i∂ s g ( v, ∇ ∂ s h ) ∇ k − i∂ s | c ′ | = k X i =0 (cid:18)(cid:18) ki (cid:19) − (cid:18) ki + 1 (cid:19)(cid:19) ∇ i∂ s g ( v, ∇ ∂ s h ) ∇ k − i∂ s | c ′ | , where we use the convention (cid:0) kk +1 (cid:1) = 0. This can be easily proved byinduction using (3.6). From this it follows that(C.1) (cid:12)(cid:12)(cid:12) D c,h ( ∇ k∂ s | c ′ | ) (cid:12)(cid:12)(cid:12) . k X i =0 (cid:12)(cid:12) ∇ i∂ s g ( v, ∇ ∂ s h ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ k − i∂ s | c ′ | (cid:12)(cid:12)(cid:12) . k X i =0 i X j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ i − j +1 ∂ s h (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ k − i∂ s | c ′ | (cid:12)(cid:12)(cid:12) , where the constant depends only on the indices i, j, k . Using the inductionhypothesis, we obtain (using the fact that | v | = 1), (cid:12)(cid:12)(cid:12) D c,h ( ∇ k∂ s | c ′ | ) (cid:12)(cid:12)(cid:12) . |∇ ∂ s h | (cid:12)(cid:12)(cid:12) ∇ k∂ s | c ′ | (cid:12)(cid:12)(cid:12) + k X i =0 i X j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ i − j +1 ∂ s h (cid:12)(cid:12)(cid:12) on every metric ball. Our assumption (5.2) implies that for i = 1 , . . . , n − k∇ i∂ s h k L ∞ ≤ C k h k G c on every metric ball. Therefore, we obtain,as long as k ≤ n − (cid:12)(cid:12)(cid:12) D c,h ( ∇ k∂ s | c ′ | ) (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) ∇ k∂ s | c ′ | (cid:12)(cid:12)(cid:12) + k X j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) k h k G c on every metric ball.In order to complete the proof (for the L ∞ case), we need to show that k∇ k∂ s v k L ∞ k = 0 , . . . , n − is bounded on every metric ball. The case k = 0 is trivial, since | v | = 1 bydefinition. Note that D c,h |∇ k∂ s v | = g ∇ h ∇ k∂ s v, ∇ k∂ s v |∇ k∂ s v | ! ≤ |∇ h ∇ k∂ s v | . Therefore, in order to use Lemma 5.7 for the function |∇ k∂ s v | , we need toshow that |∇ h ∇ k∂ s v | ≤ C (1 + k∇ k∂ s v k ∞ ) k h k G c on every metric ball. Using (3.9), we obtain ∇ h ∇ k∂ s v = ∇ ∂ s ∇ h ∇ k − ∂ s v − g ( v, ∇ ∂ s h ) ∇ k∂ s v + R ( v, h ) ∇ k − ∂ s v = ∇ k∂ s ∇ h v − k − X i =0 ∇ i∂ s ( g ( v, ∇ ∂ s h ) ∇ k − i∂ s v ) + k − X i =0 ∇ i∂ s ( R ( v, h ) ∇ k − − i∂ s v )= ∇ k +1 ∂ s h − k X i =0 ∇ i∂ s ( g ( v, ∇ ∂ s h ) ∇ k − i∂ s v ) + k − X i =0 ∇ i∂ s ( R ( v, h ) ∇ k − − i∂ s v ) , where in the last line we used the fact that ∇ h v = ∇ ∂ s h − g ( v, ∇ ∂ s h ) v, which follows immediately from (3.6). We therefore have, ∇ h ∇ k∂ s v = ∇ k +1 ∂ s h − k X i =0 i X j =0 j X l =0 (cid:18) ij (cid:19)(cid:18) jl (cid:19) g ( ∇ l∂ s v, ∇ j − l +1 ∂ s h ) ∇ k − j∂ s v + k − X i =0 i X j =0 j X l =0 l X m =0 (cid:18) ij (cid:19)(cid:18) jl (cid:19)(cid:18) lm (cid:19) ∇ j − l∂ s R ( ∇ m∂ s v, ∇ l − m∂ s h ) ∇ k − − j∂ s v, where we repeatedly used ∇ ∂ s ( R ( X, Y ) Z ) = ( ∇ ∂ s R )( X, Y ) Z + R ( ∇ ∂ s X, Y ) Z + R ( X, ∇ ∂ s Y ) Z + R ( X, Y ) ∇ ∂ s Z. Using the fact that ∇ r∂ s R is bounded for every r , we obtain the bound |∇ h ∇ k∂ s v | . |∇ k +1 ∂ s h | + k X j =0 j X l =0 |∇ l∂ s v | |∇ j − l +1 ∂ s h | |∇ k − j∂ s v | (C.3) + k − X j =0 |∇ k − − j∂ s v | j X l =0 l X m =0 |∇ m∂ s v | |∇ l − m∂ s h | (C.4) . |∇ k +1 ∂ s h | + |∇ k∂ s v ||∇ ∂ s h | + k X i =0 P i |∇ i∂ s h | , (C.5) Note that by Lemma 5.11, the whole analysis here is done on a compact subset of N (the closure of the image of B ( c , r )). Hence the boundedness of R and its covariantderivatives follows from the smoothness of N , and does not require any global boundedgeometry assumption on N (except from completeness). OBOLEV METRICS ON SPACES OF MANIFOLD VALUED CURVES 45 where P i are polynomials in |∇ ∂ s v | , . . . , |∇ k − ∂ s v | . The induction hypothesisis k∇ j∂ s v k ∞ is bounded on metric balls for j = 0 , . . . , k −
1, hence P i isbounded on metric balls. Using, this, and assumption (5.2), we obtain that,as long as k ≤ n − |∇ h ∇ k∂ s v | . (1 + |∇ k∂ s v | ) k h k G c , which completes the proof of (C.2) and hence of (5.6).It remains to prove (5.7) for k = n −
1, that is, to prove that k D c,h ( ∇ n − ∂ s | c ′ | ) k L ≤ C (1 + k∇ n − ∂ s | c ′ |k L ) k h k G c . Using (C.1) we have (cid:12)(cid:12) D c,h ( ∇ n − ∂ s | c ′ | ) (cid:12)(cid:12) . n − X i =0 i X j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ∇ i − j +1 ∂ s h (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ n − − i∂ s | c ′ | (cid:12)(cid:12) . (cid:12)(cid:12) ∇ n∂ s h (cid:12)(cid:12) | c ′ | + k h k G c n − X i =0 i X j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ n − − i∂ s | c ′ | (cid:12)(cid:12) . (cid:12)(cid:12) ∇ n∂ s h (cid:12)(cid:12) | c ′ | + k h k G c (cid:12)(cid:12) ∇ n − ∂ s | c ′ | (cid:12)(cid:12) + | c ′ | (cid:12)(cid:12) ∇ n − ∂ s v (cid:12)(cid:12) + n − X i,j =0 (cid:12)(cid:12)(cid:12) ∇ j∂ s v (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) ∇ i∂ s | c ′ | (cid:12)(cid:12) . (cid:12)(cid:12) ∇ n∂ s h (cid:12)(cid:12) + k h k G c (cid:0)(cid:12)(cid:12) ∇ n − ∂ s | c ′ | (cid:12)(cid:12) + (cid:12)(cid:12) ∇ n − ∂ s v (cid:12)(cid:12) + 1 (cid:1) where in the second inequality we used (5.2), and in the bounds (5.6) and(C.2) on metric balls. Squaring this and integrating, we obtain, using (5.3)for the first term, k D c,h ( ∇ n − ∂ s | c ′ | ) k L ≤ C (1 + k|∇ n − ∂ s v |k L + k∇ n − ∂ s | c ′ |k L ) k h k G c . Therefore, we are left to show that k|∇ n − ∂ s v |k L is bounded on metric balls.As before, we need to show that k∇ h ∇ n − ∂ s v k L ≤ C (1 + k∇ n − ∂ s v k L ) k h k G c , (C.6)and we have shown that |∇ h ∇ n − ∂ s v | . |∇ n∂ s h | + |∇ n − ∂ s v ||∇ ∂ s h | + n − X i =0 P i |∇ i∂ s h | where P i are polynomials in |∇ ∂ s v | , . . . , |∇ n − ∂ s v | , which are bounded on met-ric balls. We therefore have, using (5.2) that |∇ h ∇ n − ∂ s v | . |∇ n∂ s h | + k h k G c (cid:0) |∇ n − ∂ s v | (cid:1) . Squaring, integrating and using (5.3), we obtain (C.6), which completes theproof.
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Martin Bauer: Department of Mathematics, Florida State University
E-mail address : [email protected] Cy Maor: Einstein Institute of Mathematics, The Hebrew University ofJerusalem
E-mail address : [email protected] Peter W. Michor: Faculty for Mathematics, University of Vienna
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