Uniqueness and Pseudolocality Theorems of the Mean Curvature Flow

Mean curvature flow evolves isometrically immersed base Riemannian manifolds M in the direction of their mean curvature in an ambient manifold M̄ . We consider the classical solutions to the mean curvature flow. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete noncompact isometrically immersed hypersurfaces M (uniformly local lipschitz) in Euclidean space, the short time existence was established by Ecker and Huisken in [9]. The short time existence and the uniqueness of the solutions to the mean curvature flow of complete isometrically immersed manifolds of arbitrary codimensions in the Euclidean space are still open questions. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. More precisely, let (M̄, ḡ) be a complete Riemannian manifold of dimension n̄ such that the curvature and its covariant derivatives up to order 2 are bounded and the injectivity radius is bounded from below by a positive constant, we prove that the solution of the mean curvature flow with bounded second fundamental form on an isometrically immersed manifold M (may be high codimension) is unique. In the second part of the paper, inspired by the Ricci flow, we prove the pseudolocality theorem of mean curvature flow. As a consequence, we obtain the strong uniqueness theorem, which removes the boundedness assumption of the second fundamental form of the solution in the uniqueness theorem. 1 ∗The research partially supported by FANEDD 200216 and NSFC 10401042. AMS Mathematics Subject Classification Numbers: Primary 53c44; secondary 35k55.