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Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.


Introduction
Given an immersion F 0 : Σ → M, the mean curvature flow (MCF) starting at F 0 is a family of immersions which moves along the mean curvature vector. The MCF is the negative gradient flow of the area functional, and has been studied extensively for the past 40 years.
When Σ is compact, the MCF starting at an immersion F 0 always exists and is unique up to a maximal time interval [0, T ). On the other hand, if Σ is non-compact, the general existence and uniqueness problem is still not solved.
The first existence result in the non-compact setting is by Ecker and Huisken [8], where they prove the existence of MCF starting at a hypersurface M 0 in R n+1 with uniform Lipschitz bound. If M 0 is an entire graph, they also show the long time existence when M 0 is merely locally Lipschitz.
We remark that the minimal Lipschitz cone constructed by Lawson and Osserman [17] may serve as an obstruction to the apriori estimates in [8] in higher codimension. Existence of non-compact MCF has only been obtained for entire graph with assumptions on smallness of Lipschitz norms [3], [4], [11], [24], [20].
Next we discuss the uniqueness of MCF. Koch and Lamm show uniqueness of MCF [11] for entire graph with small Lipschitz bound in any codimension. Chen and Peng prove in [7] that any viscosity solution of the graphical Lagrangian MCF with a continuous initial data is unique. For general immersions, Chen and Yin show in [5] the uniqueness of MCF among flows with uniformly bounded second fundamental forms. Together with a pseudolocality theorem, they prove uniqueness of MCF starting from an proper embedding with bounded second fundamental form and satisfying an uniform graphic condition.
The first goal of this paper is to prove the following uniqueness theorem which generalizes Chen and Yin's uniqueness result to the case of possibly unbounded curvatures. (1) F, F are uniformly continuous with respect to t, (2) The induced metric g(t),g(t) on Σ are uniformly equivalent to g 0 , and (3) the second fundamental forms A, A satisfy for some L > 0.
The precise definition in condition (1) and (2) is given in section 2.
If we compare the above theorem to theorem 1.1 and 1.3 in [5], we assume weaker curvature bounds, while in the expense of assuming the volume growth (1.2) of the initial immersions. We do not make any graphic/curvature assumptions on the initial immersion. We remark that condition (1) and (2) are both satisfied if |A|, | A| are uniformly bounded.
Note that in theorem 1.1 we assume F 0 is smooth and the MCF F, F are both smooth up to time t = 0. In the next theorem, it is shown that under a better bound on the second fundamental forms andR, one can relax these assumptions and drop the volume growth condition. Theorem 1.2. Let (M, h) be a non-compact complete Riemannian manifold with positive injectivity radius lower bound and uniformly bound on∇ iR for i = 0, 1, 2. Let F 0 : Σ → M be a C 3 proper immersion and let F, F : (0, T ] × Σ → M be solutions to the MCF so that F, F converges to F 0 locally in C 3 as t → 0. If the curvatures satisfy |A|+| A| ≤ L/t α for some α < 1/2 and L > 0. Then F = F . The proof of the above theorems, like all other uniqueness results in MCF, use the parabolicity of the MCF equation. The technical issue is that the equation is not strictly parabolic -it's invariant under diffeomorphisms. In the previous approaches [5], [3], [4], [11], [24], [20], they use the well-known De Turck trick to construct a family of diffeomorphisms so that the resulting equation (Mean curvature De Turck flow) becomes strictly parabolic (note that the use of De Turck tricks are implicit in the graphical case, see for example p.548-549 in [8]).
Our proof of theorem 1.1 and theorem 1.2 uses directly the parabolic equation satisfied by the second fundamental form. We employ an energy argument first performed by Kotschwar in [12], where he proves a uniqueness result for non-compact Ricci flow. The energy argument was then used again for other geometric flows [13], [15], [19], [1], [2], [18], [23]. The main idea is to consider the quantity where Q is some quantity so that Q = 0 implies uniqueness. The goal is to show E(t) = 0 given that E(0) = 0. For example, in the Ricci flow situation [12], Q contains a term of the form t −β |g −g| 2 for some β, where both g,g are solutions to the Ricci flow.
In our situation, we choose our Q to contain the zeroth order term As we will see later, first and second order terms should also be present in Q in order to obtain a nice differential inequality for E(t). We do not need higher order quantity though: the parabolic nature of MCF gives a nice parabolic equation for the second fundamental form A, and an integration by part give a strictly negative term containing the third order quantities ∇A, which cancels all other third order quantities. As we will see later, cutoff functions are inserted in the energy E to deal with the non-compact situation.
Let us point out one key technical difference between our works and those in [12], [13], [19], [1], [2], [18]: In their energy arguments, the flows they consider are intrinsic, as opposed to MCF which is extrinsic. Not only that both the curvatures of Σ and M play a role, but also that the geometric quantities of two a-priori different MCFs live in different vector bundles on Σ. Thus one needs to use a bundle isomorphisms P to identity these bundles before estimating the difference. In our situation, we construct P using a parallel transport along the shortest paths between two MCFs. We remark that the same construction is also carried out in [23], [21] in the context of Schrödinger flow. We expect the same argument should work for other extrinsic geometric flows.
As a by-product, we obtain the following uniqueness result for Ricci flow, which generalizes results in [6], [12]. Theorem 1.3. Let (M, g 0 ) be a smooth complete noncompact Riemannian manifold. Let g(t),g(t), t ∈ [0, T ] be two smooth complete solutions to Ricci flow with initial metric g 0 . Suppose that g(t),g(t) are uniformly equivalent to g 0 and The last main result is the following backward uniqueness theorem for MCF. The proof of theorem 1.4 uses again that the second fundamental form A and its derivatives ∇A both satisfy strictly parabolic equations. While the lower order quantities do not, one can show that they satisfy an ordinary differential inequality. These coupled inequalities are sufficient to show theorem 1.4 by a general backward uniqueness theorem in [16]. The reader may find more historical remarks in the introduction of [16].
One slight technical issue is that the distance (1.4) is non-differentiable when it's zero and thus we need another zeroth order quantity. We treat F as a graph of F and represent F by a section on the pullback bundle F −1 T M. The assumptions on the fourth covariant derivatives ofR in theorem 1.4 is used in estimating the parabolic equation for ∇A, which we do not need in the proof of theorem 1.1.
In section 2, we fix the notations and prove some elementary results. The parallel transport P will be studied in section 3 and 4. The main estimates are performed in section 5. Theorem 1.1, theorem 1.2 and theorem 1.3 are proved in section 6 and theorem 1.4 is proved in section 7.
Acknowledgement The first author would like to thank Professor Luen-Fai Tam for his constant support and encouragement. The second author would like to thank Professor Jingyi Chen for the discussions and pointing out the reference [23]. Part of the works was done when the second author visited The Chinese University of Hong Kong and he would like to thank Professor Martin Li for the hospitality.

Prelminary and notations
In this section, we review some definitions and results in basic submanifold theory and MCF. Let Σ be a smooth manifold and (M, h) be a smooth Riemannian manifold. Let Next we introduce several notations. We write only the notations for F . A tilde will be added to the corresponding notations for F . We use (x 1 , · · · x n ) and (y 1 , · · · , y N ) respectively to denote the local coordinates on Σ, M. We use i, j, · · · to denote the indices of Σ, α, β, · · · , and α ′ , β ′ · · · respectively to denote the indices on F −1 T M, F −1 T M. For each t, let g = g(t) = F (t, ·) * h and ∇ be the Levi-Civita connection with respect to g. We use the same notation ∇ to denote the induced connection on all (p, q)-tensor bundle T p,q Σ.
We say that the families of immersions F is uniformly continuous with respect to t, if for all δ > 0 and s ∈ [0, T ), there is s δ > 0 so that We say that the induced metric g(t) is uniformly equivalent to g 0 = F * 0 h, if there is λ > 1 so that On the pullback bundle N := F −1 T M we have the connection induced from h, F : Here Γ α βγ denote the Christoffel symbols of the Levi-Civita connection∇ on (M, h). Note that Γ α βγ is indeed Γ α βγ • F , but we suppress F for simplicity. We also remark that where F i = ∂ i F and Y is any extension of Y in M. We use the same notation ∇ F to denote the connection induced by ∇ and ∇ F on any N-valued tensor bundle. Thus there could be six notations in total: However, for simplicity we use only ∇ and ∇. It will be clear from the context which connection we are using. Next we consider covariant derivatives with respect to time. Define the covariant time Note that when acts on vector fields along F , D t is metric with respect to h. That is, Next we introduce several standard geometric quantities from an immersion. For each fixed t, the differential of The mean curvature vector H is the trace of A given by Remark 1. In this paper we use the following convention: We use B k (resp. B loc k ) to denote the (resp. local) bound on |∇ kR |. Unless otherwise specified, we use C to denote constants that depend only on the dimensions of Σ and M, constants λ in (2.2), L in the statement of theorem 1.1 and the lower bound on injectivity radius i 0 . Constants that depend also on B 0 , B 1 , · · · , B k (resp. B loc 0 , B loc 1 , · · · , B loc k ) are denoted C k (resp. C loc k ). The explicit values of the constants C, C k , C loc k are not important and might change from line to line unless otherwise specified.
The following simple lemma is used a lot in this paper.
Next we recall the differential equations of the following quantities along the MCF, the proof can be found in [22].
We also need the equation for the higher covariant derivatives of A. Recall that in the notation of Chen and Yin [5], we have F * = ∇F, A = ∇ 2 F, ∇A = ∇ 3 F and so on. Proposition 2.3 in [5] together with Gauss equation give where a = 2 or 4, b = 1 or 2, * are any contraction of tensors andR include any contraction of the Riemann curvature tensor on M with h −1 .
In [5], the authors derive an apriori estimates for |∇ k F | 2 assuming that the second fundamental form ∇ 2 F is uniformly bounded. When the second fundamental form is bounded by Lr 2−ǫ /t, one can modify the proof of theorem 3.2 in [5] to obtain the following lemma.
Sketch of proof. From proposition 3.1 in [5], we have Using the conditions on |∇ 2R | and |A|, we have Thus one can proceed as in the proof of theorem 3.2 in [5] to conclude.

Writing F as a graph of F : Basic estimates
In this section, we represent F as a graph of F and provide some basic estimates. Let . We assume that Since d < i 0 , for all (t, x) there is a unique shortest geodesic joining p andp. Write exp p (sv), where s ∈ [0, 1], to denote this geodesic. Thenp = exp p v, |v| = d and v is a smooth section on N. The collection of these geodesics form a smooth homotopy We useγ, J t and J i to denote the derivative of γ with respect to s, t, and the coordinate x i respectively (the notations are so chosen since J t , J i are Jacobi fields). Note also that v, F t and F i are the restrictions ofγ, J t , J i to s = 0 respectively. Let P : TpM → T p M be the parallel transport along the geodesic −γ. The inverse P −1 is the parallel transport along γ.
On the endomorphism bundle End( N , N) = N * ⊗ N over Σ, there is a connection ∇ induced from ∇ F and ∇ F . Thus for any endomorphism valued (p, q)-tensor bundle there are two connections ∇ and ∇ induced from g andg respectively. Note that the connections satisfy the Leibniz rule: In the following we derive estimates for v and d. Since the calculations might be useful for other geometric situations, we do not assume that F, F satisfy the MCF equation except for theorem 3.1 in this section. We remark that all of the estimates follow from the Jacobi field equation (and its higher order derivatives).
First we prove a useful lemma.
, be the unique geodesic joining p(τ ) top(τ ). Let J be the Jacobi field given by the variation of geodesics γ τ . Then Proof. The first inequality (3.2) follows from the second one using |J| ≤ |∂ τ p|+sup |∇˙γJ|. To show (3.3), note that J satisfies the Jacobi field equation By the mean value theorem, there is ζ ∈ [0, 1] so that Thus by mean value theorem again, for any s ∈ [0, 1], there is ζ s between ζ and s so that where we have used (3.4) and Note that in lemma 3.1 we assume that d > 0. Indeed the Jacobi field is not even defined at points where d = 0. Thus in the MCF setting, the lemma cannot be applied directly when e.g. t = 0. To deal with this, we first consider the case d > 0 and conclude by letting d → 0 (See the proof of proposition 3.1 for an example).
The first application of lemma 3.1 is the following proposition.
Proposition 3.1. Let F, F be two families of immersions so that (3.1) holds. Then we have Thus (3.7), (3.8) follows from lemma 3.1 when d > 0. Assume d = 0 at some (t, x). Let (t i , x i ) be a sequence so that (t i , x i ) → (t, x) and d(t i , x i ) > 0. Since D t v, ∇v are smooth, the two inequalities can be shown by taking i → ∞. If such a sequence does not exists, then v is identically zero in a space time neighborhood and so D t v = ∇v = 0.
Since |v| = d, proposition 3.1 gives an estimate for ∂ t d 2 . We also use the following lemma which can also be proved easily (See Lemma 2.2 in [23]).

Lemma 3.2. We have
Next we need the following generalization of lemma 3.1. First we need a definition.

11)
where P 1 k , Q 1 k are universal polynomials in (k + 1)-variables. Proof. We argue by induction. The case k = 1 is shown using (3.3) with (3.11) holds for all integers strictly smaller than k . First we see that with i r ≤ k for all r and Using the induction hypothesis (note k q , l q ≤ k − 1) and Cauchy Schwarz inequality, one easily obtains |∇˙γ∇˙γ∇ k J J| ≤ C loc k d 2 P k−1 (|∂ τ p|, · · · , |D k−1 τ ∂ τ p|) where P k−1 , Q k−1 are universal polynomials in k-variables. Now, as in the proof of Lemma 3.1, we use mean value theorem twice to obtain This finishes the proof of the proposition.

By proposition 3.2 and |∇
kp J J| ≤ |D kp τ ∂ τ p| + sup |∇˙γ∇ kp J J|, one obtains (3.14). An immediate consequence is the following theorem, which says that if F and F are both MCF starting at F 0 , then they agree infinitesimally. Proof. Note that when v = 0 at (t, x), write p(τ ) = F (τ, x) andp(τ ) = F (τ, x), then where J is the variational vector field of the family of geodesic joining p top. From proposition 3.3, we have . Since both F and F satisfy the MCF equation, D k t H and D k t H satisfy the same parabolic equation and thus |P D k t H − D k t H| = 0 at t = 0 for all k. Hence D k t v| t=0 = 0 is shown by an approximation argument.

Estimates for the parallel transport P
Next we estimate the norm of the parallel transport P . We follow the same notations as in the previous section .First we prove the following lemma.  Proof. Since g = h(F * , F * ), together with Lemma 2.1, Using the same argument and H = g ij A ij , Thus (4.2) follows from (4.1). Proof. Since P is given by P (X ⊗Z) = X ⊗P Z for all X ∈ Γ(Σ, T p,q Σ) and Z ∈ Γ(Σ, N ), one can without loss of generality assume that p = q = 0.
Since parallel transport preserves length, |P Z| = |Z| for all Z ∈ TpM and thus |P | = |P −1 | = 1. To show (4.4), note again it suffices to assume d = 0. Let x = (x 1 , · · · , x n ) ∈ Σ, i ∈ {1, · · · , n} and Z ∈ TpM be fixed. Parallel transport Z along the curve in M with tangent vector F i . Thus ∇ i Z = 0 and by the Leibniz rule, Note that ∇ i (P Z) = (∇ J i X)(0), where X is the parallel vector field along γ with X(1) = Z. Differentiating the parallel transport equation gives Together with lemma 2.1 this implies |(∇ i P )Z| = |∇ i (P Z)| ≤ C loc 0 d|Z|. Since this holds for all Z, we obtain (4.4). To show (4.5), we calculate under the normal coordinate at x in (Σ, g t ). Thus ∆P = i ∇ i ∇ i P . Now for each fixed i and Z ∈ TpM, we again parallel transport Z along the curve in M with tangent vector F i . Thus where again X is the parallel vector field along γ with X(1) = Z. Sincē On the other hand, if we trace the induction proof of proposition 3.2 and use (4.2), we have and this gives (4.5).
The proof of the following proposition is similar to that of proposition 4.1 and is skipped. |D t P | ≤ C loc 0 d(|F t | + |P F t − F t |). Next we derive an estimates for higher time covariant derivatives of P .
Proof. As in the proof of proposition 4.1, let Z ∈ TpM and extend it to a parallel vector fields along the integral curve of F t . Then , where X is the parallel transport of Z along γ. We will prove by induction that where P 3 k−1 , Q 3 k−1 are universal polynomials of k-variables. When k = 1, one can show as in the proof of proposition 4.1 the following estimates: (4.11) |∇ Jt X| ≤ C loc 0 d|F t | + C|P F t − F t | |Z|. Next we assume that (4.10) holds for all integers strictly smaller then k. by commuting covariant derivatives, we havē Thus the induction step is proved using |∇ k Jt J t | ≤ |D t F t |+sup |∇˙γ∇ k Jt J t |, proposition 3.2, proposition 3.3, the induction hypothesis and Cauchy Schwarz inequality. This finishes the proof of the proposition.
Next, we prove the following lemma which estimates the difference of the restriction of ambient tensor to F and F . Let S be a (p, q)-tensor on M. Then S| F is a section of the bundle N ⊗p ⊗ (N * ) ⊗q over Σ. Let P * (S| F ) be given by for all a i ∈ N * , b j ∈ N.

Main estimates
In this section we provide the necessary estimates in the next two sections. In this section, we assume that F, F are both solutions to the MCF starting at F 0 which satisfies (2.2) and (2.1). In particular, by choosing a small T , we assume that d satisfies (3.1).
First we estimate the time derivative of the quantities P F * − F and Γ − Γ.
To estimate the time derivative of Γ − Γ. From (2.8) we have Thus |∂ t (Γ − Γ)| can be estimated as in the proof of Lemma 5.1 and (4.1). We skip the proof of the following lemma: Next let us consider the second order quantity Proof. From (5.3) and (2.6) and proposition 4.2, Now use (2.9) with k = 2 to write Usingh( A, A) = h(P A, P A), one has (I) = (g −1 −g −1 ) * g −1 * h(P A, P A) * P A Similarly we have For (III), note Thus a similar calculation and lemma 4.2 give Similar for (IV ) we have

Now (5.4) is obtained using Cauchy Schwarz inequalities.
6. Proof of theorem 1.1, theorem 1.2 and theorem 1.3 In this section, we use the energy argument to prove the theorem 1.1, theorem 1.2 and theorem 1.3.
Here a and b is some constant to be fixed later. Now for t ∈ (0, a/2b], define the energy E r (t) as Note that the above is well defined, since φ is of compact support, while (2.1) and the properness of F 0 together imply that F (t, ·) is also proper. From lemma 6.1 and ∂ t dµ ≤ 0 by (2.7), We focus on the term containing the Laplacians. Using (4.4), lemma 2.3 and Cauchy Schwarz inequality, Now we use integration by part to the second term on the right hand side to obtain For the first two terms, we use agaiñ proposition 4.1 and Cauchy Schwarz inequality to get For (C) we have similarly Combine all these, where A y 0 (r/2, r) ⊂ M is the annulus centered at y 0 andC is a fixed constants depending only on the dimensions of Σ, M, λ and L. To estimate the last term on the right hand side of (6.1), note Now we require that a, b satisfy a −1 ≥ 2D and b/(8nλ) ≥C. Therefore, the differential inequality for the energy quantity reduces to For each r >> 1, from now on we consider C 1 , C as fixed constants and write α = C 1 r 2−ǫ > 0. Solve the above ode on 0 < s < t < a/(2b): At each r >> 1, choose p = 2(2 + α) = 2(2 + C 1 r 2−ǫ ). For r large (depending only on C, D, ǫ, a) we have e r 2 /(4a) > D( √ Cp) p . Hence, for large enough r. By theorem 4.1 and the MCF equation, since the convergence F (t, ·), F (t, ·) → F 0 (·) are smooth, Q (m) (0) = 0 for any m ∈ N. Since F 0 is proper and M is complete, F −1 0 (B M (y 0 , 2r)) is a compact set and we may apply the dominated convergent theorem to conclude that Proof of theorem 1.2. In this situation, we observe that conditions (1) and (2) in theorem 1.1 hold since t −2α is integrable in [0, T ]. Thus all the calculations in the sections 3, 4, and 5 can be applied. Let In this situation, one use an intrinsic cutoff function: Let ρ = ρ T ∈ C ∞ (Σ) be an exhaustion of (Σ, g T ) so that for some x 0 ∈ Σ, Let φ, η be defined as in the proof of theorem 1.1 with this new ρ and let The assumption |A| + | A| ≤ C/t α implies the estimates |∇A| + | ∇ A| ≤ C 1 /t 2α . Arguing as in the proof of theorem 1.1, we have for r >> 1 and b/(8nλ) ≥C, Using the assumption on |A|, | A|, (2.2) and pick p = 2, we have Since the convergence F (t, ·), F (t, ·) → F 0 (·) is C 3 , E r (t) is continous at t = 0 and E r (0) = 0. Integrating the above inequality (note t −2α is integrable) gives Let r → ∞ gives Q α = 0 for all t ∈ [0, a/2b]. Thus F = F in [0, a/2b] and the theorem follows from iterating the argument.
Using the above cutoff technique and the argument in the proof of theorem 1.1, we sketch how one can prove theorem 1.3.
Sketch of proof of theorem 1.3. We argue using similar argument in [12]. Define the energy to be Here we choose the cutoff function and exhaustion function as in the proof of theorem 1.1: φ(x) = φ(ρ(x)/R) and η(x, t) = ρ(x) 2 a−bt where ρ is a smooth function on M such that d 0 (x, x 0 ) ≤ ρ(x) ≤ d 0 (x, x 0 ) + 1 and |∇ g 0 ρ| ≤ 2 for some x 0 ∈ M. By volume comparison and equivalence of metrics, we know that Using integration by part, we obtain a evolution inequality of E R which is in the same form as before.
for some L = L(n, λ). We can now employ the same trick in the proof of theorem 1.1 to conclude that g(t) =g(t) for all t ∈ [0, T ].

Backward Uniqueness
In this section, we modify a general backward uniqueness result in [16] to prove theorem 1.4. A similar approach was carried out in [10], where they prove a backward uniqueness theorem for MCF in the co-dimension one case. Unfortunately, the issue of parallel transport is not addressed in [10] when the ambient space is not Euclidean.
We use the metric induced from g and h and the direct sums are orthogonal. Define the following time covariant derivatives on X and Y respecitvely: Consider the following two sections X, Y on X , Y respectively: where v is defined in section 3. Theorem 1.4 follows from the following Proof. First we estimate ∂ t (∇(Γ − Γ)). We remark that for any (p, q) tensors S on Σ, we have (∂ t ∇ − ∇∂ t )S = (∂ t Γ) * S.
To estimate the third term we use From (2.9), one can check that Using the above inequalities with the case k = 0, 1 give (7.1) and the theorem is proved.
Proof of theorem 1.4. In [16], the author proves a general backward uniqueness theorem for two sections X, Y in two fixed vector bundles X , Y on Σ respectively. We remark that their proof goes through if one assume that X , Y are both time dependent vector bundle with ∂ t replaced by D X t , D Y t . In particular, to apply theorem 3 in [16] to our situation, let τ = T − t, Λ ij = g ij .