Local existence and uniqueness of Skew Mean Curvature Flow
aa r X i v : . [ m a t h . DG ] A p r LOCAL EXISTENCE AND UNIQUENESS OF SKEW MEAN CURVATUREFLOW
CHONG SONG
Abstract.
The Skew Mean Curvature Flow(SMCF) is a Schr¨odinger-type geometric flow canon-ically defined on a co-dimension two submanifold, which generalizes the famous vortex filamentequation in fluid dynamics. In this paper, we prove the local existence and uniqueness of generaldimensional SMCF in Euclidean spaces.
Contents
1. Introduction 11.1. Definition and backgrounds 11.2. Main results 31.3. Further problems 4Acknowledgement 52. Local existence of SMCF 52.1. Idea of proof 52.2. Uniform Sobolev interpolation inequalities 52.3. Evolution equations 62.4. Proof of existence 83. Uniqueness of SMCF 93.1. Idea of proof 93.2. Parallel transport and distance of tensors 113.3. Bundle isomorphisms between two solutions 133.4. Difference of Laplacian operators 153.5. Estimate of L Introduction
Definition and backgrounds.
The Skew Mean Curvature Flow(SMCF) is a Schr¨odingertype geometric flow defined on a co-dimension two submanifold, which evolves the submanifoldalong its bi-normal direction with the speed given by its mean curvature. More specifically, suppose( ¯
M , ¯ g ) is an ( n + 2) dimensional oriented Riemannian manifold, Σ is an n dimensional orientedmanifold, then the SMCF is a family of time-dependent immersions F : [0 , T ) × Σ → ¯ M satisfying(1.1) ∂ t F = J ( F ) H ( F ) . Here H ( F ) is the mean curvature vector field of the submanifold and J ( F ) is the induced complexstructure on its normal bundle N Σ. Namely, J rotates a normal vector ν by π/ Date : April 9, 2019. ormal plane, such that for any oriented basis { ǫ , · · · , ǫ n } ⊂ T x Σ, F ∗ ( ǫ ) ∧ F ∗ ( ǫ ) ∧ · · · ∧ F ∗ ( ǫ n ) ∧ ν ∧ ( J ν ) coincides with the orientation of ¯ M .The simplest model of SMCF is the one dimensional SMCF in three dimensional Euclidean space,which reduces to the famous Vortex Filament Equation(VFE) ∂ t γ = γ s × γ ss = κ b . Thus the SMCF is a geometric generalization of the VFE both in higher dimensions and in generalambient Riemannian manifolds. It naturally arises in both the context of superfluid [Jer02] andclassical hydrodynamics [Sha12, Khe12], describing the locally-induced motion of a co-dimensiontwo vortex membrane. For a detailed introduction to the backgrounds and motivations to studythe SMCF, we refer to our previous paper [SS19].The SMCF falls in the category of dispersive, or more precisely, Schr¨odinger type geometricflows. Probably due to lack of analytic tools, the research of Schr¨odinger type geometric flows arerather underdeveloped as compared to their parabolic (or hyperbolic) counterparts. Little is knownabout the SMCF except for the 1 dimensional VFE, which bears very rich structures.It is well-known that the VFE is equivalent to a standard cubic Schr¨odinger equation by theHasimoto transformation, hence is completely integrable and behaves nicely. For example, theinitial value problem of VFE is globally well-posed for smooth initial curves, and has a global weaksolution which enjoys a weak-strong uniqueness property for integral currents as initial data [JS15].Similarly, the one dimensional SMCF in a general three dimensional Riemannian manifold is alsoequivalent to a Schr¨odinger equation and exists globally [Gom04]. However, higher dimensionalSMCFs ( n ≥
2) are essentially different from the VFE and such a magical transformation seemsimpossible. Very recently, Khesin and Yang [KY19] provide an evidence from the point of viewof integrable systems. They also construct an interesting example of the product of differentdimensional spheres where the SMCF blows up in finite time.From the perspective of PDE analysis, there are two main difficulties in the investigation of a highdimensional ( n ≥
2) SMCF. First of all, for a general (non-graphical) high dimensional submanifold,there does not exist a global coordinate system or a global gauge on its normal bundle. Thusmany powerful analytical tools in the study of dispersive equations, such as harmonic analysis, areinvalid. Secondly, even if we write down the equation in local coordinates, the SMCF still seemsvery challenging. In fact, it is well-known that the mean curvature H ( F ) is a quasi-linear anddegenerate elliptic operator on F . In local coordinates H ( F ) can be written as H ( F ) α = (∆ g F ) α = g ij ( ∂ i ∂ j F α − Γ kij ∂ k F α + ¯Γ αβγ ∂ i F β ∂ j F γ ) , where g ij = ¯ g ( ∂ i F, ∂ j F ) is the induced metric, Γ kij and ¯Γ αβγ are the Christoffel symbols of (Σ , g )and ( ¯ M , ¯ g ) respectively. In particular, Γ kij involve with the first order derivatives of g ij and hencesecond order derivatives of F . Moreover, the complex structure J ( F ) also involves with the firstorder derivatives of F . Thus the high dimensional SMCF (1.1) can be regarded as a quasi-linearSchr¨odinger type system that is defined on a manifold.As far as the author knows, there are only a few results on the existence and uniqueness of generalquasi-linear Schr¨odinger equations on Euclidean spaces, usually under strong restrictions on thenon-linear structures. See for example [KPV04], the recent book [LP15] and references therein.On the other hand, during the last twenty-years, there has been remarkable developments onthe research of another typical Schr¨odinger type flow, namely, the celebrated Schr¨odinger mapflow. Suppose ( M, g ) is a Riemannian manifold and (
N, ω ) is a symplectic manifold with an almostcomplex structure J N , then the Schr¨odinger map flow u : [0 , T ) × M → N is defined by the equation(1.2) ∂ t u = J N ( u ) τ ( u ) , here τ ( u ) is the tension field of u . The local well-posedness, finite time blow-up and global behaviorof Schr¨odinger map flow are extensively studied, see for example [Din02, RRS09, BIKT11, MRR13,Li18] and references therein. Compared to the SMCF, the Schr¨odinger map flow is considerablyeasier since the corresponding operator τ ( u ) is semi-linear and elliptic. However, there is a strongconnection between the SMCF and the Schr¨odinger map flow.In [Son17], we found that the Gauss map of a SMCF satisfies a Schr¨odinger map flow into aGrassmannian manifold with evolving metric. Indeed, the Gauss map of a SMCF in an Euclideanspace is a map ρ : [0 , T ) × (Σ , g ) → G ( n,
2) mapping to the Grassmannian manifold G ( n, G ( n,
2) by J , then ρ satisfies(1.3) ∂ t ρ = J ( ρ ) τ g ( ρ ) . The key difference between (1.3) and ordinary Schr¨odinger map flow (1.2) is that now the metric g of the underlying manifold is also evolving along time, by the equation ∂ t g = − h J H, A i , Thus the SMCF can be regarded as a coupled Schr¨odinger map flow, which is more sophisticated,and new methods must be deployed.In [SS19], the author and Sun proved the local existence of two dimensional SMCF in R byapplying a parabolic regularization and an energy method. The proof essentially relies on anuniform estimate of the second fundamental form of a two dimensional surface, which is obtainedby using blow-up techniques in geometric analysis. However, it seems that the key estimate onlyholds for two dimensional surfaces. In this paper, we will prove both local existence and uniquenessof SMCF in general dimensions.1.2. Main results.
Suppose Σ is an n -dimensional compact oriented manifold where n ≥ F : Σ → R n +2 , we consider the initial value problem(1.4) (cid:26) ∂ t F = J H,F (0 , · ) = F ( · ) . Let k = [ n ] + 1 where [ n ] denotes the integer part of n/
2. Fix a small number δ ∈ (0 ,
1) and let p = n + δ . For k ≥ k , we define the energy of an immersion F by E k ( F ) = vol + k H k p + k A k k, , where k A k k, is the H k, -Sobolev norm with regard to the normal connection ∇ on the productbundle N Σ ⊗ ( T ∗ Σ) s , namely, k A k k, := k A k H k, = k X l =0 Z |∇ l A | ! / . Our main result is the following theorem.
Theorem 1.1.
Suppose Σ is an n -dimensional compact oriented manifold with n ≥ and F :Σ → R n +2 is a smooth immersion, then the initial value problem (1.4) has a unique smooth solution F : [0 , T ) × Σ → R n +2 , where T only depends on the energy E k ( F ) of the initial immersion F .Moreover, there exists a constant C k only depending on k and E k ( F ) such that for all t ∈ [0 , T ) , (1.5) E k ( F ( t )) ≤ C k E k ( F ) , ∀ k ≥ k . Remark 1.2. (1)
With some efforts, one can generalize the above results to SMCFs into gen-eral ambient Riemannian manifolds with bounded geometry. But here we assume the ambi-ent space is Euclidean space for simplicity. In the appendix, we show that the energy E k is equivalent to the W k +1 , -Sobolev norm ofthe Gauss map. See Theorem 4.2 for more details. (3) In [DW01] , Ding and Wang obtained the existence of a local solution to the Schr¨odingermap flow with W k +1 , -initial value, which seems to be optimal. In view of the fact thatthe Gauss map ρ of the SMCF satisfies a Schr¨odinger map flow (1.3), our estimate (1.5)which only depends on E k , or equivalently on the W k +1 , -norm of ρ , is consistent withDing-Wang’s result. (4) It is very tempting to prove the existence of a strong solution F ∈ L ∞ ([0 , T ) , W k +2 , ) tothe SMCF with an initial immersion F ∈ W k +2 , . However, this can not be achieved byour estimate (1.5). Because the W k, - bound of the second fundamental form A dose notdirectly yield a W k +2 , -bound on the immersion F . There is a loss of derivative due tothe diffeomorphism group of the domain manifold. Actually, we can only get a solution F ∈ L ∞ ([0 , T ) , W k +1 , ) in this case. The existence part of Theorem 1.1(see Theorem 2.12 below) is proved in Section 2 by a parabolicregularization method. Namely, we first solve a perturbed SMCF which is parabolic, and then tryto find a limit solution to the SMCF by letting the perturbation vanish. The choice of the energy E k is crucial in the proof which yields a uniform estimate for the perturbed SMCF. It is natural toput the norms of the second fundamental form k A k k, in the energy E k , while the term vol + k H k p is enclosed to ensure that the Sobolev constants in the interpolation inequalities are uniform. Formore explanation of the strategy of proof, see Section 2.1 below.The uniqueness of the SMCF in Theorem 1.1 follows from the following more general theorem.To state the result, we define the k -th space of immersions for k ∈ N by S k := { F : Σ → R n +2 | F is an immersion with k A k k, ∞ < ∞} . Theorem 1.3.
Suppose Σ is an n -dimensional compact oriented manifold with n ≥ and F :Σ → R n +2 is an immersion in S . If F ∈ L ∞ ([0 , T ] , S ) and ˜ F ∈ L ∞ ([0 , T ] , S ) are twosolutions of the SMCF (1.4) with same initial value F , then F = ˜ F a.e. on [0 , T ] × Σ . Remark 1.4.
Note that here we only requires one of the solutions lie in the space L ∞ ([0 , T ] , S ) while the other solution in a weaker space L ∞ ([0 , T ] , S ) . This can be compared with the weak-stronguniqueness of the VFE, i.e. one dimensional SMCF, proved by Jerrard and Smets [JS15] . The proof of Theorem 1.3, which is presented in Section 3, turns out to more involved than that ofthe existence. Our proof of uniqueness of the SMCF is based on an energy method which originatesfrom our previous work on the uniqueness of the Schr¨odinger map flow [SW18]. The main idea issimply to define an energy functional L = L ( F, ˜ F ) which describes the distance of two solutions,and try to show that L vanishes identically in the time span [0 , T ]. However, finding such a suitablefunctional L takes a lot of efforts and requires a full exploration of the geometric structure of theSMCF. In our final solution, we define L by using parallel transport on the Grassmannian manifold.It is worth mentioning that during the proof, we propose a notion of intrinsic distance between thesecond fundamental forms of two submanifolds, which might be of independent interest. A detailedexposition of the idea behind the construction of L is given in Section 3.1.1.3. Further problems.
Finally, we propose several open problems on the well-posedness of theSMCF:(1) Prove the local existence and uniqueness of a weak(or strong) solution of the SMCF formore general initial submanifolds that are less regular. This might involve a well-definednotion of weak solutions of the SMCF, cf. [Jer02].
2) Prove the existence of SMCF for complete non-compact submanifolds under suitable as-sumptions. In particular, we are interested in the existence of a local solution to the SMCFdefined on R n .(3) Is the SMCF globally well-posed for initial submanifolds with certain small energy? Acknowledgement
Part of this work was carried out during a visit at Tsinghua University from September 2017 toFebruary 2018. The author would like to thank Professor Yuxiang Li, Huaiyu Jian and HongyanTang for their support and hospitality. He is also grateful to Professor Yu Yuan, Jingyi Chen andYoude Wang for stimulating conversations and their generous help over the past several years.2.
Local existence of SMCF
Idea of proof.
To prove the local existence of the SMCF, we apply an approximating schemeby considering a perturbed SMCF(2.1) ∂ t F = J ε H := J H + εH, ε > . For ε >
0, the perturbed flow (2.1) is a linear combination of the SMCF and the Mean CurvatureFlow(MCF). It turns out to be a (degenerate) parabolic system which behaves similarly as theMCF. A standard argument by using the De Turck trick guarantees the existence of a local solution F ε : [0 , T ε ) → R n +2 to (2.1) for any ε >
0. Therefore, if we can derive a uniform estimate of F ε anda positive lower bound of T ε , then we obtain a local solution to the SMCF (1.1) by letting ε → ε , especially for Schr¨odinger typegeometric flows, which are quasi-linear. Note that since the parabolic term vanishes as ε →
0, anyparabolic type estimates, including those obtained by maximum principal, would blow-up and donot yield the desired uniform bounds.Nevertheless, we are still able to derive a uniform bound of the second fundamental form A ε corresponding to F ε by using an energy method. Namely, we consider the evolution equation of the H k, -Sobolev norms of A ε on [0 , T ε ), and try to control the nonlinear terms by Sobolev interpolationinequalities. The main obstruction for implementing such an idea is that the Sobolev constants inthe standard interpolation inequalities for tensors are not uniform, as they depend on the underlyingmetric, which is also evolving along the flow. To overcome this problem, we will apply a uniformSobolev inequality (see Theorem 2.4 below) proved by Mantagazza [Man02], which says that theSobolev constants are uniform if vol + k H k p is uniformly bounded for some p > n . This motivatesus to enclose the term into the total energy and define E k = vol + k H k p + k A k k, . The point is that the Sobolev constants are uniform as long as the energy E k is uniformly bounded.By considering the evolution equation of E k of F ε , we manage to derive a uniform bound on E k ,which in turn gives a uniform bound of A ε . Then the convergence of F ε to a solution of the SMCFfollows by a standard argument.2.2. Uniform Sobolev interpolation inequalities.
In this subsection, we recall several uniformSobolev interpolation inequalities which will be used later.First we recall the following standard universal interpolation inequality.
Theorem 2.1 (Aubin [Aub98], Chapter 3, Section 7.6) . Suppose M is a compact m -dimensionalRiemannian manifold. Let E be a vector bundle on M , which is endowed with a metric and acompatible connection D . Then for any section s ∈ Γ( E ⊗ ( T ∗ M ) p ) and exponents q ∈ [1 , ∞ ) and ∈ [1 , ∞ ] , there is a constant C only depending on the dimension m and the exponents such thatfor all < j < k k D j s k p ≤ C k D k s k jk q k s k − jk r , where kp = jq + k − jr Remark 2.2.
The above theorem also applies for complete non-compact manifolds (cf. [Can75] ). Next we have the famous Michael-Simon inequality for submanifolds.
Theorem 2.3 (Michael-Simon [MS73]) . Let M is an immersed compact m -dimensional subman-ifold in the Euclidean space with mean curvature H . Then for any smooth function u : M → R and p ∈ [1 , m ) , we have k u k p ∗ ≤ C ( k Du k p + k Hu k p ) , where C is a constant only depending on the dimension m and the exponent p . The following theorem is a quite straight forward application of Theorem 2.3, and provides theuniform interpolation theorem which is used in this paper. The proof involves first proving (2.2)for a = 1 , j = 0 , k = 1 by applying Theorem 2.4, and then following a standard procedure as in theproof of classical Gagliardo-Nirenberg inequality. Theorem 2.4 (Mantegazza [Man02]) . Suppose M is an m -dimensional compact immersed sub-manifold of the Euclidean space R n . If V ol + k H k n + δ ≤ B for some δ > , then we have uniformGagliardo-Nirenberg inequality for any covariant tensor T . In particular, there is a constant C onlydepending on B, m and the exponents such that (2.2) k D j T k p ≤ C k T k ak,q k T k − ar , where j ∈ [0 , k ] , p, q, r ∈ [1 , ∞ ] and a ∈ [ j/k, ( a = 1 if q = m/ ( k − j ) = 1 ) satisfies p = jm + a (cid:18) q − km (cid:19) + 1 − ar > . In particular, when kq > m , we have k T k ∞ ≤ C k T k W k,q . Evolution equations.
For a small number ε ≥
0, suppose F : [0 , T ] × Σ → R n +2 is a solutionto the perturbed SMCF(2.3) ∂ t F = J ε H = εH + J H, where J ε = εI + J and H is the mean curvature vector.First we recall some evolution equations under the perturbed SMCF, which are obtained inSong-Sun [SS19]. Note that when ε = 0, they reduce to evolution equations of the SMCF. Lemma 2.5.
Under the perturbed SMCF (2.3), we have the following evolution equations ∂ t g = − h J ε H, A i , (2.4) ∂ t dµ = − ε | H | dµ, (2.5) ∂ t A = J ε ∆ A + A A A, (2.6) ∂ t H = J ε ∆ H + A A H, (2.7) ∂ t ∇ l A = J ε ∆ ∇ l A + X i + j + k = l ∇ i A ∇ j A ∇ l A. (2.8) here g is the induced metric, A is the second fundamental form, dµ is the induced volume form, ∇ and ∆ is the normal connection and corresponding Laplacian operator, and denotes linearcombinations of tensors. The Gauss map of F is a map ρ : [0 , T ] × (Σ , g ) → G into the Grassmannian manifold G = G ( n, Lemma 2.6.
Under the perturbed SMCF (2.3), the Gauss map ρ satisfies ∂ t ρ = J ε ( ρ ) τ g ( ρ ) = ετ g ( ρ ) + J ( ρ ) τ g ( ρ ) , where J is the complex structure on G and τ g ( ρ ) is the tension field of ρ . Next we derive the evolution equation of the Sobolev norms of A . Since by integration by parts, Z D ∇ l A, J ε ∆ ∇ l A E = − ε Z |∇ l +1 A | , it follows from (2.8) that ∂ t k∇ l A k ≤ C X i + j + k = l Z M |∇ i A | · |∇ j A | · |∇ k A | · |∇ l A | dµ. Applying H¨older’s inequality and the universal interpolation inequality in Theorem 2.1 with r = ∞ ,we get Lemma 2.7.
Under the perturbed SMCF (2.3), ∂ t k∇ l A k ≤ C k A k ∞ k∇ l A k . Hence (2.9) ∂ t k A k k, ≤ C k A k ∞ k A k k, . We will also need the evolution equation of the L p -norm of H . Lemma 2.8.
Under the perturbed SMCF (2.3), for p ≥ , (2.10) ∂ t k H k p ≤ C ( k∇ H k p + k A k ∞ k H k p ) . Proof.
By direct computation, ∂ t k H k pp = Z p | H | p − h H, ∇ t H i = p Z | H | p − h H, J ǫ ∆ H + A A H i = − p ( p − Z | H | p − h H, ∇ H i h H, J ǫ ∇ H i − p Z | H | p − h∇ H, J ǫ ∇ H i + p Z | H | p − h H, A A H i≤ p ( p − Z |∇ H | | H | p − + p Z | A | | H | p ≤ C ( k∇ H k p k H k p − p + k A k ∞ k H k pp ) . Thus the lemma follows. (cid:3)
Remark 2.9.
Up to now, we have only used the universal interpolation inequality in Theorem 2.1,thus all the constants are uniform. .4. Proof of existence.
Let k = [ n , p = 2 nn − k + 2 . Note that k > n/
2, thus we have Sobolev embedding W k , ֒ → L ∞ . Moreover p > n and wehave Sobolev embedding W k , ֒ → W ,p for all n < p < p . (Actually, p = 2 n when n is odd, and p = + ∞ when n is even.)Now let p ∈ ( n, p ) be a fixed real number. For the perturbed SMCF with ε >
0, we define theenergy E k ( F ε ) := vol + k H k p + k A k k, , where vol is the volume and k ≥ k is an integer.First we assume that k H k p is uniformly bounded along the perturbed SMCF for some time span[0 , T ε ]. Recall that by (2.5), the volume is decreasing along the perturbed SMCF. Thus there existsa uniform constant B > k H k p ≤ B for t ∈ [0 , T ′ ε ].Then we can apply Theorem 2.4 to get k A k ∞ + k∇ A k p ≤ C ( B ) k A k k , . It follows immediately from inequality (2.9) that ∂ t k A k k, ≤ C ( B ) k A k k , k A k k, . Similarly, by (2.10) we have ∂ t k H k p ≤ C ( B ) k A k k , (1 + k H k p ) . Therefore, we conclude that
Lemma 2.10.
Suppose there is a constant
B > , such that vol + k H k p ≤ B along the perturbedSMCF for a time span [0 , T ′ ε ] , then for any k ≥ k , (2.11) ∂ t E k ≤ C ( B ) k A k k , · (1 + E k ) . The following lemma shows that there is a uniform lower bound for the time T ′ ε for ε > Lemma 2.11.
Given an initial immersion F : M → R m +2 , there exists a uniform time T > only depending on E := E k ( F ) such that E k ( t ) := E k ( F ε ( t )) ≤ E + 1 , ∀ ε > , t ∈ [0 , T ] . Proof.
Set B = 2(1 + E ). For each ε >
0, define T ′ ε := sup { T ∈ [0 , T ε ] | E k ( t ) ≤ B, ∀ t ∈ [0 , T ] } . Clearly, T ′ ε > F ε is smooth. Moreover, T ′ ε is smaller than the maximal existencetime T ε . Otherwise we can find a limit of F ε ( t ) as t → T ε and the solution can be extended pastthe maximal time T ε .Applying Lemma 2.10, we get that for t ∈ [0 , T ′ ε ] ∂ t (1 + E k ) ≤ C ( B ) k A k k , (1 + E k ) ≤ ¯ C ( B )(1 + E k ) , where ¯ C ( B ) = C ( B ) B is a constant only depending on B .It follows from Gronwall’s inequality that1 + E k ( t ) ≤ e ¯ C ( B ) t (1 + E ) . Now by letting t = T ′ ε , we find 2 ≤ e ¯ C ( B ) T ′ ε , yielding T ′ ε ≥ T := ln 2¯ C ( B ) . (cid:3) ith the above uniform estimates, we are now in the position to prove the existence part ofTheorem 1.1, which we restate as follows. Theorem 2.12.
Suppose Σ is a compact n dimensional manifold and F is a smooth immersion,then the initial value problem (1.4) has a smooth solution F : [0 , T ) × Σ → R n +2 , where T onlydepends on the energy E = E k ( F ) , k = [ n/
2] + 1 .Moreover, there exists a constant C k only depending on k and E such that for all t ∈ [0 , T ) , (2.12) E k ( F ( t )) ≤ C k E k ( F ) , ∀ k ≥ k . Proof.
For any small ε >
0, by applying DeTurck’s trick, we can find a local solution F ε : [0 , T ǫ ] × M → R m +2 to the perturbed SMCF (2.3). (cf. Lemma 4.1 in [SS19])By Lemma 2.11, there exists a time T only depending on E such that T ε > T and(2.13) E k ( F ε ( t )) ≤ E + 1 , ∀ ε > , t ∈ [0 , T ] . Then by (2.11) in Lemma 2.10 that for all ε >
0, we have uniform inequality ∂ t E k ( F ε ( t )) ≤ C ( E )(1 + E k ( F ε ( t ))) , which implies(2.14) E k ( F ε ( t )) ≤ C ( E ) E k ( F )) . In particular, we have(2.15) k A ( F ε ( t )) k k, ≤ C k ( E ) k A ( F ) k k, , ∀ k ≥ k . Since vol + k H k p of each submanifold F ε ( t ) are uniformly bounded by (2.13), we have uniformSobolev inequalities guaranteed by Theorem 2.4. Thus it follows from (2.15) that the secondfundamental form A ( F ε ( t )) and its derivatives are all uniform bounded.Now by standard arguments, we can extract a subsequence ε i → F ε i convergessmoothly to a limit F ∞ : [0 , T ] × M → R m +2 which is the desired solution to the SMCF. Finally,the estimate (2.12) is a direct consequence of (2.14). (cid:3) Remark 2.13.
By Theorem 4.2 in the appendix, the energy E k is equivalent to the energy ¯ E k = k ρ k W k +1 , of the Gauss map. The latter energy is used by Mantegazza [Man02] to investigate ahigher order analog of the mean curvature flow. Uniqueness of SMCF
Idea of proof.
The proof of uniqueness of the SMCF is more challenging than that of exis-tence. In fact, it is well-known that even for a (complex-valued) scalar Schr¨odinger equation thatcontains 1st order derivatives (also referred as derivative Schr¨odinger equations), the uniquenesscan only be obtained under certain restrictions on the nonlinear structure (cf. [LP15]). Comparedto parabolic equations, this is because the leading 2nd order term in a Schr¨odinger equation doesnot play a dominant role and provide a “good” term which help to control the lower-order terms.Nevertheless, by fully exploring the underlying geometric structures of the SMCF, we show thatthe uniqueness still holds true, provided the solution is sufficiently smooth.Suppose F and ˜ F are two solutions to the initial value problem (1.4) with the same initial data F , we will show that the two solutions are identical by an energy method. Thus we need to definean energy functional L which describes the difference/distance of F and ˜ F such that L vanishes ifand only if F = ˜ F . The main problem then is to find a suitable quantity L .One naive option is to take the distance of two submanifolds ¯ d ( F, ˜ F ) and its derivatives, where ¯ d is the extrinsic distance in the ambient space ¯ M . In our setting where ¯ M = R n +2 , one may simplylet L = k F − ˜ F k k, for some k ∈ N . But the evolution equations of derivatives of F and ˜ F are quite essy, because they are not geometric quantities (tensors) of corresponding submanifolds. Thusthis method certainly won’t work and we need to find more geometric quantities instead.The most important geometric quantity of a submanifold is the second fundamental form. Thusone may consider the difference of the second fundamental forms, such as k A − ˜ A k . Note that A is a section of the bundle N ⊗ ( T ∗ Σ) while ˜ A is a section of the bundle ˜ N ⊗ ( T ∗ ˜Σ) , where N and ˜ N are the normal bundle of Σ = (Σ , g ) and ˜Σ = (Σ , ˜ g ) respectively. Thus by defining thesubstraction A − ˜ A , we implicitly use the embedding N ⊂ F ∗ R n +2 , ˜ N ⊂ ˜ F ∗ R n +2 and the distanceof the ambient space R n +2 . Moreover, we directly identity T ∗ Σ with T ∗ ˜Σ although their metricsare different.However, by the evolution equation (2.6), we have ∂ t ( A − ˜ A ) = J ∆ A − ˜ J ˜∆ ˜ A + A − ˜ A . The leading order terms on the right hand side will cause serious technical issues. First of all, itwon’t make sense if we write J ∆ A − ˜ J ˜∆ ˜ A = J (∆ A − ˜∆ ˜ A ) + ( J − ˜ J ) ˜∆ ˜ A, because the complex structure J and ˜ J are only defined on corresponding normal bundles. Even ifwe insist to do so by extending the definition of J and ˜ J , some first order terms will emerge fromthe term ∆ A − ˜∆ ˜ A , which seems hard to control. Moreover, it is useless to add higher order termsinto the energy L , say k∇ A − ˜ ∇ ˜ A k , because yet an even higher order term will appear and theissue remains unsolved.From the above discussion, one can see that the difficulty of proving uniqueness is caused by thehigher order terms which appear while we take the difference of tensors defined on two differentsubmanifolds. Our solution is to find a more geometric way to measure the difference of two sub-manifolds. In particular, we will define bundle isomorphisms between tensor bundles on differencesubmanifolds, to eliminate the extra terms appearing during the substraction. Our key idea comesfrom the observation that the Gauss map of the SMCF satisfies a Schr¨odinger map flow [Son17]and our investigation on uniqueness of Schr¨odinger map flows [SW18].Given two solutions F, ˜ F of the SMCF, there are two Gauss maps ρ, ˜ ρ which satisfies a Schr¨odingermap flow (with time-dependent metrics on the domain manifold) in the Grassmannian G . Thenfollowing [SW18], we define the intrinsic distance of dρ and d ˜ ρ by the parallel transport P on theGrassmannian manifold. But dρ and d ˜ ρ can be identified with the second fundamental forms A, ˜ A respectively. Therefore, we are led to define the intrinsic distance of A and ˜ A by d ( A, ˜ A ) = d ( dρ, d ˜ ρ ) = | dρ − P ( d ˜ ρ ) | . In other words, the parallel transport map P induces a bundle isomorphism between ρ ∗ T G = N ⊗ T ∗ Σ and ˜ ρ ∗ T G = ˜
N ⊗ T ∗ ˜Σ, such that J ◦ P = P ◦ ˜ J . This solves the issue brought by theextrinsic method above. Namely, now we have J ∆ A − P ( ˜ J ˜∆ ˜ A ) = J (∆ A − P ( ˜∆ ˜ A )) . Actually in our proof, we go one step further and define a bundle isometry R s : ˜ N ⊗ ( T ∗ ˜Σ) s →N ⊗ ( T ∗ Σ) s for any s ∈ N . Then we consider the energy L ( F, ˜ F ) = k d ( ρ, ˜ ρ ) k + k A − R ( ˜ A ) k + k∇ A − R ( ˜ ∇ ˜ A ) k . It turns out that using R s instead of P can help us further reduce the requirements on the regularityof the solutions.On the other hand, we also need to estimate the difference of the two Laplacian operatiors ∆and ˜∆. Since the underlying metrics g and ˜ g of the two solutions are different, we shall encode the ifference of the metrics and connections into the total energy by letting L ( F, ˜ F ) = k g − ˜ g k + k Γ − ˜Γ k . Moreover, because L is defined via the parallel transport while L is defined by directly identi-fying T Σ with T ˜Σ, we also need to fill the gap between L and L . This is accomplished by addinga term L ( F, ˜ F ) = k I − Qk , where Q : T ˜Σ → T Σ is the isometry induced by P and I : T ˜Σ → T Σ is the identity map.Finally, we define the total energy functional by L = L + L + L . Then the uniqueness ofSMCF follows from a Gronwall type inequality of L . Remark 3.1. In [CY07] and [LM17] , the authors proved the uniqueness of the Mean CurvatureFlow(MCF) by using the parallel transport P ¯ M in the ambient space ¯ M . It should by pointed out thatour parallel transport P G in the Grassmannian manifold is more sophisticated than P ¯ M . Indeed, inour setting where ¯ M = R n +2 is the Euclidean space, the parallel transport P ¯ M is simply the identitymap, while P G is highly non-trivial. More precisely, P ¯ M provides an isomorphism between thepull-back bundles F ∗ T ¯ M and ˜ F ∗ T ¯ M , but P G establishes an isomorphism between both the normalbundles N , ˜ N and the tangent bundles T Σ , T ˜Σ . Parallel transport and distance of tensors.
Intrinsic distance of vectors.
Suppose M is a Riemannian manifold. Let x, y ∈ M be twodistinct points and X ∈ T x M , Y ∈ T y M be two tangent vectors. How can we define the distancebetween X and Y ?Note that in principal X and Y are vectors in two difference tangent spaces, so it does not makesense to do subtraction directly. However, if we embed M into an ambient Euclidean space by ι : M → R K , then we can think of X and Y as vectors in R k and define d ( X, Y ) := | dι x ( X ) − dι y ( Y ) | R K . We shall call d the extrinsic distance of tangent vectors. The disadvantage of this distance isobvious: it relies on the embedding ι and there is no canonical embedding.So we seek for more intrinsic distances. One natural way is to use the parallel transport to definean isomorphism between different tangent spaces. More precisely, suppose there is a geodesic γ : [0 , → M connecting x and y , then we can parallel transport a vector along γ , which give riseto an isomorphism P : T y M → T x M . Next we define d ( X, Y ) := | X − P ( Y ) | x . This distance is intrinsic but in general depends on the choice of geodesic γ . Fortunately, if x and y lies close enough, then we have a unique minimizing geodesic and the distance d is canonicallydefined.The intrinsic distance d is particularly useful in proving uniqueness. Moreover, d naturallyarises when we take derivatives of the distance function on the manifold. Indeed, if d : M × M → R is the distance function on M , then ∇ ( X,Y ) d ( x, y ) = (cid:10) − γ ′ (0) , X (cid:11) x + (cid:10) γ ′ (1) , Y (cid:11) y = (cid:10) − γ ′ (0) , X − P ( Y ) (cid:11) x . In fact, the intrinsic distance d defined by the parallel transport can be generalized to any vectorbundle. Suppose π : ( E, ∇ , h ) → M is a vector bundle over M . X ∈ E x , Y ∈ E y are two vectorsat x, y ∈ M which are connected by a geodesic γ . Thus we can find a lift V : [0 , → E of γ suchthat ∇ s V = 0 and V (1) = Y . The parallel transport P : E y → E x is then obtained by assigning P ( Y ) = V (0), and the distance is defined by d ( X, Y ) = | X − P ( Y ) | x . emark 3.2. There is still another intrinsic way to define the distance by using Jacobi fields.Namely, suppose there is a Jacobi fields V on the geodesic γ such that V (0) = X and V (1) = Y ,then we define d ( X, Y ) := (cid:18)Z |∇ s V | ds (cid:19) . It turns out that this distance is equivalent to d defined above, see for example [CJW13] . Remark 3.3. In [DW98] , Ding-Wang proved the uniqueness of Schr¨odinger map flow for C solutions using the extrinsic distance d . In [McG07] , McGahagan showed that the uniquenessholds in a larger function space by using the intrinsic distance d . This result was improved bySong-Wang [SW18] using the same idea but more intrinsic methods. Parallel transport on Grassmannian manifolds and intrinsic distance of second fundamentalforms.
A key idea of the current paper is to define an intrinsic distance of tensors by paralleltransport in vector bundles over Grassmannian manifolds, which enable us to compare the secondfundamental forms of two submanifolds in a more geometric way. For a preliminary introductionto the geometry of Grassmannian manifolds, we refer to [Son17].First recall that there are two canonical bundles on the Grassmannian manifold G , namely thetautological bundle G ⊤ where the fiber G ⊤ ξ at each ξ ∈ G is ξ itself, and the normal counterpart G ⊥ where the fiber G ⊥ ξ at ξ is its normal complement space ξ ⊥ . Moreover, the tangent bundle G := T G is isomorphic to the product bundle G ⊤ ⊗ G ⊥ .Now suppose F, ˜ F : Σ → R K are two submanifolds, and ρ, ˜ ρ : Σ → G are their Gauss maps.For each x ∈ Σ , let γ x : [0 , → G be a geodesic in G connecting ρ ( x ) and ˜ ρ ( x ). Then by paralleltransport in the bundles G ⊤ , G ⊥ , G , we have three isomorphisms P ⊤ x : G ⊤ ˜ ρ ( x ) → G ⊤ ρ ( x ) , P ⊥ x : G ⊥ ˜ ρ ( x ) → G ⊥ ρ ( x ) , P x : G ˜ ρ ( x ) → G ρ ( x ) . In particular, P x = P ⊤ x ⊗ P ⊥ x .Next we let x vary on Σ, and suppose there is a family of geodesics γ x connection ρ ( x ) and ˜ ρ ( x )for every x ∈ Σ. Then we get three bundle isometries P ⊤ : ˜ ρ ∗ G ⊤ → ρ ∗ G ⊤ , P ⊥ : ˜ ρ ∗ G ⊥ → ρ ∗ G ⊥ , P : ˜ ρ ∗ G → ρ ∗ G . Again, we have P = P ⊤ ⊗ P ⊥ .Recall by definition, G ⊤ ρ ( x ) = ρ ( x ) = T F ( x ) M is the tangent plane of M = F (Σ ) at F ( x ), and G ⊥ ρ ( x ) = ρ ( x ) ⊥ = N F ( x ) M is the normal plane at F ( x ). Thus we can identify the bundles H := F ∗ T M ∼ = ρ ∗ G ⊤ , N := F ∗ N M ∼ = ρ ∗ G ⊥ . Similarly, ˜ H := ˜ F ∗ T ˜ M ∼ = ˜ ρ ∗ G ⊤ , ˜ N := ˜ F ∗ N ˜ M ∼ = ˜ ρ ∗ G ⊥ . Therefore, the above bundle isomorphisms constructed by the parallel transport actually give ustwo bundle isometries between the tangent bundles and normal bundles, respectively, i.e. P ⊤ : ˜ H → H , P ⊥ : ˜ N → N . Finally, to extend the parallel transport to tensors including the second fundamental form A ∈ Γ( N ⊗ T ∗ Σ ⊗ T ∗ Σ), we only need to find a bundle isometry between T ∗ Σ and H . This can be easilyaccomplished as follows.First note that the tangent map dF gives a bundle isometry between T Σ := ( T Σ , g ) and H .Similarly, d ˜ F gives a bundle isometry between T ˜Σ := ( T Σ , ˜ g ) and ˜ H . This in turn gives an sometry Q between the tangent bundles T Σ and T ˜Σ by letting the following diagram commute: T ˜Σ Q (cid:15) (cid:15) d ˜ F / / ˜ H P ⊤ (cid:15) (cid:15) T Σ dF / / H In other words, we define Q := dF − ◦ P ⊤ ◦ d ˜ F : T ˜Σ → T ΣSince d ˜ F , P ⊤ , dF are all isometries, Q is obviously an isometry.Next we may identify the cotangent bundle T ∗ Σ (or T ∗ ˜Σ) with the tangent bundle T Σ (respec-tively T ˜Σ) by sending an orthonormal frame { ǫ ∗ , ǫ ∗ , · · · , ǫ ∗ n } to its dual { ǫ , ǫ , · · · , ǫ n } . Thus byusing the above isometry Q , we have a dual isometry on the cotangent bundles Q ∗ : T ∗ ˜Σ → T ∗ Σ.Now for any s ∈ N , we can define a bundle isometry by R s := P ⊥ ⊗ ( Q ∗ ) s : ˜ N ⊗ ( T ∗ ˜Σ) s → N ⊗ ( T ∗ Σ) s . which gives an intrinsic distance of tensors T ∈ Γ( N ⊗ ( T ∗ Σ) s ) , ˜ T ∈ Γ( ˜
N ⊗ ( T ∗ ˜Σ) s ) by d ( T, ˜ T ) = | T − R s ( ˜ T ) | g . In particular, we can define the “intrinsic distance” of the second fundamental forms by d ( A, ˜ A ) = | A − R ( ˜ A ) | g . Bundle isomorphisms between two solutions.
Construction of Bundle isomorphisms.
Now suppose F and ˜ F are two solutions to the SMCF(1.1) with same initial data F . Then there Gauss maps ρ and ˜ ρ are two solutions to (1.3) withsame initial data ρ . We assume that their is a time T > ρ and ˜ ρ lies sufficiently closeto each other such that for any ( t, x ) ∈ [0 , T ] × Σ, there is a unique geodesic γ ( t,x ) : [0 , → G connecting ρ ( t, x ) and ˜ ρ ( t, x ). The family of geodesics γ ( t,x ) gives rise to a map U (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [0 , × [0 , T ] × Σ → G ( s, t, x ) γ ( t,x ) ( s )By definition, we have U (0 , t, x ) = ρ ( t, x ) and U (1 , t, x ) = ˜ ρ ( t, x ). Note that the family of geodesicsdepends smoothly on its end points given by ρ and ˜ ρ .As in [SW18], we have the following bound for the derivatives of U . Lemma 3.4.
There holds |∇ t U | ≤ C ( | ∂ t ρ | + | ∂ t ˜ ρ | ) ≤ C ( |∇ ρ | + | ˜ ∇ ˜ ρ | ) , |∇ U | ≤ C ( |∇ ρ | + | ˜ ∇ ˜ ρ | ) , |∇ U | ≤ C ( |∇ ρ | + | ˜ ∇ ˜ ρ | ) + C ( |∇ ρ | + | ˜ ∇ ˜ ρ | ) . Moreover, (cid:18)Z |∇ ∂ s U | ds (cid:19) ≤ |∇ ρ − P ( ˜ ∇ ˜ ρ ) | + C ( |∇ ρ | + | ˜ ∇ ˜ ρ | ) d. From previous discussion in Section 3.2, we know that by parallel transport along U , we havebundle isomorphisms P ⊤ : ˜ ρ ∗ G ⊤ → ρ ∗ G ⊤ , P ⊥ : ˜ ρ ∗ G ⊥ → ρ ∗ G ⊥ , P = P ⊤ ⊗ P ⊥ : ˜ ρ ∗ G → ρ ∗ G . n particular, P ⊤ induces bundle isomorphisms between tangent bundles and cotangent bundles Q = dF − ◦ P ⊤ ◦ d ˜ F : T ˜Σ → T Σ , Q ∗ : T ∗ ˜Σ → T ∗ Σ . Moreover, we have an extended bundle isomorphism for tensor bundles(3.1) R s := P ⊥ ⊗ ( Q ∗ ) s : ˜ N ⊗ ( T ∗ ˜Σ) s → N ⊗ ( T ∗ Σ) s . Estimate of derivatives of P . By construction, the bundle isomorphism P preserves thebundle metric. Moreover, since the complex structure on G is parallel, P also preserves the complexstructure of the bundle, i.e. P ◦ ˜ J = J ◦ P . However, the two pull-back connections ∇ = ρ ∗ ∇ G and˜ ∇ = ˜ ρ ∗ ∇ G does not commute with P . Actually, there holds¯ ∇P = ∇ ◦ P − P ◦ ˜ ∇ . Here we regard P as a tensor in Γ( ˜ G ∗ ⊗ G ) = Hom( ˜ G , G ) and ¯ ∇ denotes the induced connection.To estimate the derivatives of P , we will use the method of moving fames to express P moreexplicitly. An alternative method is to use the Jacobi equation, which is applied in [LM17].For fixed ( t, x ) ∈ I × Σ, we first choose an orthonormal frame { E a } of ρ ∗ G near ρ ( t, x ), andthen parallel transport it along the connecting geodesics to get a local frame { ¯ E a ( s ) } of U ∗ G over γ ( t,x ) ( s ). Denote the resulting frame at ˜ ρ ( t, x ) by { ˜ E a := ¯ E a (1) } , which is also orthonormal. Weshall call the chosen parallel orthonormal frame on the bundles by relative frame .Obviously, in this frame, we have P ( ˜ E a ) = E a . Thus by letting ˜Ω a := ˜ E ∗ a be the dual frame, wecan simply write P = ˜Ω a ⊗ E a ∈ Γ( ˜ G ∗ ⊗ G ) . Now in the relative frame, the connections has the form ∇ = d + A and ˜ ∇ = d + ˜ A , where A = A ba,i dx i and ˜ A = ˜ A ba,i dx i are connection 1-forms. Then we compute¯ ∇ P = ˜ ∇ ˜Ω a ⊗ E a + ˜Ω a ⊗ ∇ E a = ( − ˜ A ab ˜Ω b ) ⊗ E a + ˜Ω a ⊗ ( A ba E b )= ( A ba − ˜ A ba ) ˜Ω a ⊗ E b . Let ¯ ∇ = U ∗ ∇ G = d + ¯ A be the pull-back connection on U ∗ G over [0 , × [0 , T ] × Σ, where ¯ A is the corresponding connection 1-form in the parallel frame { ¯ E a } . Since the frame we choose areparallel along s -direction, we can write ¯ A = ¯ A t dt + ¯ A i dx i . The curvature form of ¯ ∇ is given by ¯ F = d ¯ A + [ ¯ A, ¯ A ] = U ∗ R G . It follows that the ds ∧ dx i component of ¯ F is¯ F si = ∂ s ¯ A i = R G ( ∂ s U, ∇ i U ) . On the other hand,since ∇ and ˜ ∇ are the restrictions of ¯ ∇ on ρ ∗ G = U ∗ G| s =0 and ˜ ρ ∗ G = U ∗ G| s =1 ,we have A ( t, x ) = ¯ A (0 , t, x ) and ˜ A ( t, x ) = ¯ A (1 , t, x ). Hence the difference of two connections is givenby A i − ˜ A i = ¯ A i | s =0 − ¯ A i | s =1 = − Z ∂ s ¯ A i ds = − Z ¯ F is ds. Therefore, we get ¯ ∇P = − Z R G ( ∂ s U, ∇ i U ) ba ds · dx i ⊗ ˜Ω a ⊗ E b . aking another derivative, we get¯ ∇ P = − ¯ ∇ (cid:18)Z R G ( ∂ s U, ∇ i U ) ba ds · dx i ⊗ ˜Ω a ⊗ E b (cid:19) = − Z ¯ ∇ ( R G ( ∂ s U, ∇ i U ) ba ) ds · dx i ⊗ ˜Ω a ⊗ E b + Z R G ( ∂ s U, ∇ U ) ds · ¯ ∇ ( dx i ⊗ ˜Ω a ⊗ E b )= − Z ( ∇ G R G )( ∂ s U, ∇ U, ∇ U ) ds − Z R G ( ∇ ∂ s U, ∇ U ) ds − Z R G ( ∂ s U, ∇ U ) ds + Z R G ( ∂ s U, ∇ U ) ds · ( A − ˜ A ) . As a conclusion, we have
Lemma 3.5.
The derivatives of P satisfies | ¯ ∇ t P| ≤ Cd |∇ t U || ¯ ∇P| ≤ Cd |∇ U | . | ˜ ∇ P| ≤ C ( |∇ U | + |∇ U | ) d + C |∇ U ||∇ ∂ s U | , where the constant C only depends on G . Remark 3.6.
By replacing the bundle G with G ⊤ and G ⊥ in the above arguments, it is easy tosee that P ⊤ and P ⊥ satisfies same estimates. Moreover, since ¯ ∇Q = ¯ ∇P ⊤ , Q also satisfies sameestimates. It follows that for any s ∈ N , the parallel translation R s satisfies same estimates. Difference of Laplacian operators.
Difference of Laplacian with different metrics.
Let ( E , ∇ , h ) be a vector bundle over a man-ifold Σ, suppose there are two difference metric g and ˜ g on Σ. We denote by ∇ g , ∆ g and ∇ ˜ g , ∆ ˜ g the corresponding induced covariant derivatives and (rough) Laplacians. Then for any sectionΦ ∈ Γ( E ), by definition∆ g Φ − ∆ ˜ g Φ = ( g ij − ˜ g ij ) ∇ i ∇ j Φ − ( g ij Γ kij − ˜ g ij ˜Γ kij ) ∇ k Φ= ( g ij − ˜ g ij ) ∇ g,ij Φ − ˜ g ij (Γ kij − ˜Γ kij ) ∇ k Φ . Thus | ∆ g Φ − ∆ ˜ g Φ | h ≤ | g − − ˜ g − | g |∇ g Φ | h ⊕ g + | ˜ g − | g | Γ − ˜Γ | g |∇ g Φ | h ⊕ g , where we use the metric g for tensors on the right hand side.3.4.2. Difference of Laplacian with different connections.
Given two vector bundles ( E, ∇ , h ) and( ˜ E, ˜ ∇ , ˜ h ) over a Riemannian manifold (Σ , g ). Suppose there is a bundle isomorphism P : ˜ E → E which preserves the metric, then we have for any Φ ∈ Γ( ˜ E ), ∇ ( P Φ) = ( ¯ ∇P )Φ + P ( ˜ ∇ Φ) , where ¯ ∇ denotes the induced connection on ˜ E ∗ ⊗ E . It follows ∇ ( P Φ) = ∇ (( ¯ ∇P )Φ + P ( ˜ ∇ Φ)= ( ¯ ∇ P )Φ + 2( ¯ ∇P ) ˜ ∇ Φ + P ( ˜ ∇ Φ) . Taking trace, we get ∆( P Φ) − P ( ˜∆Φ) = ( ¯∆ P )Φ + 2 ¯ ∇P · ˜ ∇ Φ . Therefore, | ∆( P Φ) − P ( ˜∆Φ) | h ≤ | ¯∆ P| ¯ h | Φ | ˜ h + 2 | ¯ ∇P| ¯ h | ˜ ∇ Φ | ˜ h , where ˜ h = ˜ h ⊕ h is the induced metric on ˜ E ∗ ⊗ E . .4.3. Difference of Laplacian with different metrics and connections.
Now for two vector bundles π : ( E, ∇ , h ) → (Σ , g ) and ˜ π : ( ˜ E, ˜ ∇ , ˜ h ) → (Σ , ˜ g ), suppose there is a bundle isomorphism P : ˜ E → E which preserves the metrics on the bundle, then we have for any Φ ∈ Γ( ˜ E ), | ∆ g ( P Φ) − P ( ˜∆ ˜ g Φ) | h ≤ | ∆ g ( P Φ) − P ( ˜∆ g Φ) | h + |P ( ˜∆ g Φ) − P ( ˜∆ ˜ g Φ) | h = | ∆ g ( P Φ) − P ( ˜∆ g Φ) | h + | ˜∆ g Φ − ˜∆ ˜ g Φ | ˜ h By the estimates above, we get
Lemma 3.7.
Under above settings, we have | ∆ g ( P Φ) − P ( ˜∆ ˜ g Φ) | h ≤ | ¯ ∇P| ¯ h | ˜ ∇ ˜ g Φ | ˜ h + | ¯∆ P| ¯ h | Φ | ˜ h + | g − − ˜ g − | ˜ g | ˜ ∇ g Φ | ˜ h ⊕ ˜ g + | g − | ˜ g | Γ − ˜Γ | ˜ g | ˜ ∇ ˜ g Φ | ˜ h ⊕ ˜ g . As a conclusion of Lemma 3.7, Lemma 3.5 and Lemma 3.4, we obtain
Corollary 3.8.
For any tensor Φ ∈ Γ( ˜
N ⊗ ( T ∗ ˜Σ) s ) and the parallel transport R s defined by (3.1),if g and ˜ g are equivalent, then there holds | ∆ g ( R s Φ) − R s ( ˜∆ ˜ g Φ) | ≤ C ( d + | A − P ( ˜ A ) | + | g − ˜ g | + | Γ − ˜Γ | ) . where C depends on k ρ k , ∞ , k ˜ ρ k , ∞ and k Φ k , ∞ . In the above theorem, since the metrics g and ˜ g are assumed to be equivalent, i.e. there existsconstants such that C g ≤ ˜ g ≤ C g , we can use either one for tensors. Thus we omit the subscriptsdenoting the metrics here and subsequently.3.5. Estimate of L . By the discussions in Section 3.3 above, we have the parallel transport R s defined by (3.1). Now we define the energy functional L ( F, ˜ F ) = L + L + L where L = k d ( ρ, ˜ ρ ) k + k A − R ( ˜ A ) k + k∇ A − R ( ˜ ∇ ˜ A ) k , and L = k g − ˜ g k + k Γ − ˜Γ k , and L = k I − Qk . Here we assume the metrics g and ˜ g are equivalent, thus we may choose a fixed background metric g (e.g. the induced metric of the initial immersion F ) to define the norm in L . In the definitionof L , I is the identity map from T ˜Σ to T Σ and the norm is taken with regard to the inducedmetric ¯ g = ˜ g ⊕ g .In what follows, we will derive a Gronwall type estimate of L .3.5.1. Estimate of L . First using equation (1.3), we compute12 ddt Z d ( ρ, ˜ ρ ) = Z (cid:10) − γ ′ (0) , ∂ t ρ (cid:11) + (cid:10) γ ′ (1) , ∂ t ˜ ρ (cid:11) = Z D − γ ′ (0) , J ( ρ )∆ ρ − P ( J (˜ ρ ) ˜∆˜ ρ ) E = Z D − γ ′ (0) , J ( ρ ) tr g ∇ dρ − J ( ρ ) P (tr ˜ g ˜ ∇ d ˜ ρ ) E = Z D J ( ρ ) γ ′ (0) , tr g ( ∇ A − R ( ˜ ∇ ˜ A )) E ≤ Z d + Z |∇ A − R ( ˜ ∇ ˜ A ) | , here d := d ( ρ, ˜ ρ ).For the second term of L , recall that by Lemma 2.5, ∇ t A = J ∆ A + A , and ˜ ∇ t ˜ A = ˜ J ˜∆ ˜ A + ˜ A . It follows ∇ t A − ∇ t R ( A ) = J ∆ A + A − R ( ˜ J ˜∆ ˜ A + ˜ A ) − ¯ ∇ t R ( ˜ A )= J (∆ A − R ( ˜∆ ˜ A )) + ( A − R ( ˜ A ) ) − ¯ ∇ t R ( ˜ A )= J ∆( A − R ( ˜ A )) + J (∆( R ˜ A ) − R ( ˜∆ ˜ A )) + ( A − R ( ˜ A ) ) − ¯ ∇ t R ( ˜ A ) . Hence12 ddt Z | A − R ( ˜ A ) | g = Z D A − R ( ˜ A ) , ∇ t A − ∇ t R ( ˜ A ) E = Z D A − R ( ˜ A ) , J (∆( R ˜ A ) − R ( ˜∆ ˜ A )) + ( A − R ( ˜ A ) ) − ¯ ∇ t R ( ˜ A ) E ≤ C (cid:18)Z | A − R ( ˜ A ) | g + Z | ∆( R ˜ A ) − R ( ˜∆ ˜ A ) | + Z | ¯ ∇ t R| (cid:19) where C depends on k A k ∞ , and k ˜ A k ∞ .By Lemma 3.5 and Lemma 3.4, | ¯ ∇ t R| ≤ Cd | ∂ t U | ≤ C ( |∇ A | + | ˜ ∇ ˜ A | ) d. On the other hand, by Corollary 3.8, | ∆( R ˜ A ) − R ( ˜∆ ˜ A ) | ≤ C ( d + | A − P ( ˜ A ) | + | g − ˜ g | + | Γ − ˜Γ | ) . where C depends on k ρ k , ∞ , k ˜ ρ k , ∞ and k ˜ A k , ∞ . Therefore, we arrive at12 ddt Z | A − R ( ˜ A ) | ≤ C (cid:18)Z d + Z | A − R ( ˜ A ) | g + Z | g − ˜ g | + Z | Γ − ˜Γ | (cid:19) . where C depends on k A k , ∞ and k ˜ A k , ∞ .For the third term of L , recall that by Lemma 2.5, ∇ t ∇ A = J ∆ ∇ A + A A ∇ A, and ˜ ∇ t ˜ ∇ ˜ A = ˜ J ˜∆ ˜ ∇ ˜ A + ˜ A A ∇ ˜ A. It follows that ∇ t ∇ A − ∇ t ( R ˜ ∇ ˜ A ) = J ∆ ∇ A + A A ∇ A − R ( ˜ J ˜∆ ˜ ∇ ˜ A − ˜ A A ∇ ˜ A ) − ¯ ∇ t R ( ˜ ∇ ˜ A )= J ∆( ∇ A − R ( ˜ ∇ ˜ A )) + J (∆( R ( ˜ ∇ ˜ A )) − R ( ˜∆ ˜ ∇ ˜ A ))+ ( A A ∇ A − R ( ˜ A ) R ( ˜ A ) R ( ˜ ∇ ˜ A )) − ¯ ∇ t R ( ˜ ∇ ˜ A ) o we have 12 ddt Z |∇ A − R ( ˜ ∇ ˜ A ) | = Z D ∇ A − R ( ˜ ∇ ˜ A ) , ∇ t ∇ A − ∇ t ( R ˜ ∇ ˜ A ) E = Z D ∇ A − R ( ˜ ∇ ˜ A ) , J (∆( R ( ˜ ∇ ˜ A )) − R ( ˜∆ ˜ ∇ ˜ A )) E + Z D ∇ A − R ( ˜ ∇ ˜ A ) , ( A A ∇ A − R ( ˜ A ) R ( ˜ A ) R ( ˜ ∇ ˜ A )) − ¯ ∇ t R ( ˜ ∇ ˜ A ) E ≤ C (cid:18)Z | A − R ( ˜ A ) | g + Z |∇ A − R ( ˜ ∇ ˜ A ) | + Z | ∆( R ( ˜ ∇ ˜ A )) − R ( ˜∆ ˜ ∇ ˜ A ) | + Z | ¯ ∇ t R ( ˜ ∇ ˜ A ) | (cid:19) . Again by Corollary 3.8, we have | ∆( R ( ˜ ∇ ˜ A )) − R ( ˜∆ ˜ ∇ ˜ A ) | ≤ C ( d + | A − P ( ˜ A ) | + | g − ˜ g | + | Γ − ˜Γ | ) . where C depends on k ρ k , ∞ , k ˜ ρ k , ∞ and k ˜ ∇ ˜ A k , ∞ . Therefore, we obtain12 ddt Z |∇ A − R ( ˜ ∇ ˜ A ) | ≤ C (cid:18)Z d + Z | A − R ( ˜ A ) | g + Z |∇ A − R ( ˜ ∇ ˜ A ) | + Z | g − ˜ g | + Z | Γ − ˜Γ | (cid:19) . In conclusion, we have(3.2) ddt L ≤ C ( L + L ) . where C depends on k A k , ∞ and k ˜ A k , ∞ .3.5.2. Estimate of L . First recall that by Lemma 2.5, ∂ t g = − h J H, A i , ∂ t ˜ g = − D ˜ J ˜ H, ˜ A E . Thus ∂ t g − ∂ t ˜ g = − h J H, A i + 2 D P ⊥ ( ˜ J ˜ H ) , P ⊥ ( ˜ A ) E = − D J ( H − P ⊥ ( ˜ H )) , A E − D P ⊥ ( ˜ J ˜ H ) , A − P ⊥ ( ˜ A ) E It follows | ∂ t ( g − ˜ g ) | ≤ C | A − P ⊥ ( ˜ A ) | ≤ C ( | A − R ( ˜ A ) | + | I − Q|| ˜ A | ) . and 12 ∂ t Z | g − ˜ g | ≤ C (cid:18)Z | g − ˜ g | + Z | A − R ( ˜ A ) | + Z | I − Q| (cid:19) , where C depends on k A k ∞ and k ˜ A k ∞ .Next, recall that the evolution equation of the Christoffel symbol is ∂ t Γ = g − ∇ ∂ t g, ∂ t ˜Γ = ˜ g − ∇ ∂ t ˜ g. Moreover, ∇ ∂ t g = − ∇ h J H, A i , ˜ ∇ ∂ t ˜ g = − ∇ D ˜ J ˜ H, ˜ A E . t follows | ∂ t (Γ − ˜Γ) | = | g − ∇ ∂ t g − ˜ g − ∇ ∂ t ˜ g |≤ C ( | g − − ˜ g − | + | A − ˜ A | + |∇ A − ˜ ∇ ˜ A | ) ≤ C ( | g − − ˜ g − | + | A − R ( ˜ A ) | + |∇ A − R ( ˜ ∇ ˜ A ) | + ( | ˜ A | + | ˜ ∇ ˜ A | ) | I − Q| ) . So we have12 ddt Z | Γ − ˜Γ | = Z D Γ − ˜Γ , ∂ t (Γ − ˜Γ) E ≤ C (cid:18)Z | Γ − ˜Γ | + Z | g − ˜ g | + Z | A − R ( ˜ A ) | + Z |∇ A − R ( ˜ ∇ ˜ A ) | + Z | I − Q| (cid:19) , where C depends on k A k , ∞ and k ˜ A k , ∞ .In conclution, we have(3.3) ddt L ≤ C ( L + L + L ) . where C depends on k A k , ∞ and k ˜ A k , ∞ .3.5.3. Estimate of L . To estimate the time derivative of L , we need to compute the evolutionequation of the identity map I : T ˜Σ → T Σ. This is most easily done in natural coordinates.Denote the natural coordinates in Σ and ˜Σ by x i and ˜ x i (which is actually the same one on theparameter space Σ ). Let ∂ i = ∂∂x i and ˜ ∂ i = ∂∂ ˜ x i . In this coordinates, the identity map is simply I = d ˜ x i ⊗ ∂ i . Then we can compute¯ ∇ t I = ˜ ∇ t d ˜ x i ⊗ ∂ i + d ˜ x i ⊗ ∇ t ∂ i = (Γ itj − ˜Γ itj ) d ˜ x j ⊗ ∂ i . On the other hand, we have Γ itj = 12 g ik ∂ t g jk = − g ik h J H, A jk i , and ˜Γ itj = 12 ˜ g ik ∂ t ˜ g jk = − ˜ g ik D ˜ J ˜ H, ˜ A jk E . It follows | ¯ ∇ t I | ≤ C ( | g − ˜ g | + | A − R ( ˜ A ) | + | I − Q| ) , where C depends on k A k ∞ and k ˜ A k ∞ .From this inequality and Lemma 3.5, we can estimate12 ddt Z | I − Q| = Z (cid:10) I − Q , ¯ ∇ I − ¯ ∇Q (cid:11) ≤ Z | I − Q| + Z | ¯ ∇ I | + Z | ¯ ∇Q| ≤ C (cid:18)Z | I − Q| + Z | g − ˜ g | + Z | A − R ( ˜ A ) | + Z d (cid:19) . In conclusion, we get(3.4) ddt L ≤ C ( L + L + L ) . where C depends on k A k ∞ and k ˜ A k ∞ . .6. Proof of uniqueness.
Now we are ready to prove Theorem 1.3.
Proof of Theorem 1.3.
By assumption F ∈ L ∞ ([0 , T ] , S ) and ˜ F ∈ L ∞ ([0 , T ] , S ), the norms k A k , ∞ and k A k , ∞ are uniformly bounded for all time t ∈ [0 , T ]. It is easy to see that the inducedmetric g and ˜ g are equivalent.Moreover, the Gauss maps ρ, ˜ ρ of F, ˜ F satisfy the Schr¨odinger map flow (1.3) with same initialvalue ρ given by the Gauss map of F . Since τ g ( ρ ) = ∇ H and τ ˜ g (˜ ρ ) = ˜ ∇ ˜ H are uniformly bounded,there exists a time T ′ > x, t ) ∈ Σ × [0 , T ′ ], the image ρ ( x, t ) , ˜ ρ ( x, t ) lies ina sufficiently small geodesic ball of the Grassmannian manifold G , which is centered at ρ ( x, t ).Therefore, we can construct the family of geodesics U connecting ρ and ˜ ρ , hence the paralleltransport/bundle isomorphisms P , Q , R s as in Section 3.3.Then we can define the energy function L as in Section 3.5. Next, combining estimates (3.2),(3.3) and (3.4), we obtain ddt L ≤ C L , ∀ t ∈ [0 , T ′ ] , where C depends on k A k , ∞ and k ˜ A k , ∞ . It follows that L ( t ) ≤ e Ct L (0) for t ∈ [0 , T ′ ]. But L (0) = 0since F and ˜ F have the same initial value. Hence L vanishes identically and F = ˜ F on [0 , T ′ ].Next, starting from the time T ′ , we can repeat the above arguments to get F = ˜ F on anothertime interval [ T ′ , T ′ + T ′′ ]. Note that T ′′ can be chosen to be equal to T ′ , since it only depends on k A k , ∞ and k ˜ A k , ∞ which are uniformly bounded.Therefore, after repeating the arguments for finitely many times, we can get the uniqueness onthe whole time interval [0 , T ], which finishes the proof of the theorem. (cid:3) Appendix
Let F : Σ n → R m be a compact immersed submanifold. In this appendix, we show that the energy E k = vol + k H k p + k A k H k, is equivalent to the Sobolev norm of the Gauss map ¯ E k = k dρ k W k, ,where the Sobolev norms will be defined later. For a preliminary introduction to the geometry ofGrassmannian manifolds and Gauss maps, we refer to [Son17].Let D denote the usual connection of the exterior product space Λ := Λ n R m , which is inducedby the standard derivative on R m . Let ∇ denote the Levi-Civita connection of the Grassmannianmanifold G := G ( n, m − n ) and Π ∈ Γ( T ∗ G ⊗ T ∗ G ⊗ N G ) denote the second fundamental formof G as a submanifold in Λ. We can regard Π as an 1-form Π = Π a dy a on G , where each entryΠ a ∈ Γ( T ∗ G ⊗ N G ) is a linear map from
T G to N G .The Gauss map of F is a map ρ : Σ → G ֒ → Λ. We will still denote the pull-back connections onthe pull-back bundles ρ ∗ T Λ ⊗ ( T ∗ Σ) s and ρ ∗ T G ⊗ ( T ∗ Σ) s by D and ∇ respectively. Then applying D on dρ ∈ Γ( ρ ∗ T G ⊗ T ∗ Σ), we have
Ddρ = (
Ddρ ) ⊤ + ( Ddρ ) ⊥ = ∇ dρ + ρ ∗ Π( dρ ) , where ρ ∗ Π = Π a ∂ i ρ a dx i is the pull-back 1-form on Σ. Since we can identify dρ with the secondfundamental form A of the immersion F , we can write the above equality as DA = ∇ A + Π( ρ ) A , where D A = D ∇ A + D (Π( ρ ) A )= ( ∇ A + Π A ∇ A ) + ( D Π A + Π DA A )= ∇ A + Π A ∇ A + ( D Π + Π A . nductively, we can derive for any k ≥ D k A = ∇ k A + X J C J (Π) ∇ j A · · · ∇ j s A, or equivalently,(4.2) D k dρ = ∇ k A + X J C J (Π) ∇ j dρ · · · ∇ j s dρ, where C J (Π) is a linear combination of Π and its derivatives, and the summation is taken for allindices J = ( j , · · · , j s ) with0 ≤ j i ≤ k − X i ( j i + 1) = X i j i + s = k + 1 , Now for k ≥
1, define the Sobolev norms k A k H k, = k X l =0 Z |∇ l A | ! / , and k dρ k W k, = k X l =0 Z | D l dρ | ! / . Theorem 4.1.
For any compact immersed submanifold F : Σ n → R m and k ≥ l := [ n/ , suppose vol + k H k p ≤ B for some p > n , then there exists two constants C ( k, B ) and C ( k, B ) which onlydepend on k and B (and is independent of the submanifold), such that (4.3) k A k H k, ≤ C ( k, B ) k +1 X i =1 k dρ k iW k, , and (4.4) k dρ k W k, ≤ C ( k, B ) k +1 X i =1 k A k iH k, . Proof.
Since by assumption vol + k H k p ≤ B , we have uniform interpolation inequalities by Theo-rem 2.4. Then in view of (4.1) and (4.2), the proof follows step by step from Proposition 2.2 inDing-Wang [DW01],. (cid:3) Theorem 4.2.
For any compact immersed submanifold F : Σ n → R m and k ≥ l := [ n/ ,the energy E k = vol + k H k p + k A k H k, is equivalent to ¯ E k = k ρ k W k +1 , . Namely, there exists twofunctions f k and g k which only depend on k (and is independent of the submanifold), such that ¯ E k ≤ f k ( E k ) , E k ≤ g k ( ¯ E k ) . Proof.
First note that | ρ | = 1 since the image of ρ lies in G which is contained in the unit spherein Λ. Thus R | ρ | = vol and k ρ k W k +1 , = vol + k dρ k W k, .If we take B = E k in Theorem 4.1, then by (4.4),¯ E k = vol + k dρ k W k, ≤ vol + C ( k, B ) k +1 X i =1 k A k iH k, . So it is easy to find the desired function f k such that ¯ E k ≤ f k ( E k ). n the other hand, by letting j = 1 , k = l + 1 , r = ∞ , q = 2 in the universal interpolationinequality of Theorem 2.1, we have k A k p = k dρ k p ≤ C k D l dρ k l k ρ k − l ∞ ≤ C ¯ E l k . where p = 2( l + 1) > n . Therefore, we getvol + k H k p ≤ B ′ := ¯ E k + C ¯ E l k . Then by (4.3) in Theorem 4.1, there exists a function g k such that E k ≤ g k ( ¯ E k ). (cid:3) References [Aub98] Thierry Aubin.
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School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P.R.China.
E-mail address : [email protected]@xmu.edu.cn