Uniqueness and pseudolocality theorems of the mean curvature flow

Abstract

Mean curvature flow evolves isometrically immersed base manifolds M in the direction of their mean curvatures in an ambient manifold M . If the base manifold M is compact, the short-time existence and uniqueness of the mean curvature flow are well known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.

Convergence of Lagrangian mean curvature flow in Kähler–Einstein manifolds

“…Now we prove that there exist κ 0 , r 0 > 0 such that L t is κ 0 -noncollapsed on the scale r 0 for t ∈ [ 1 2 t 0 , t 0 ]. In fact, by Proposition 2.2 in [2] or Theorem 2.1 in [7] the injectivity radius of L is bounded from below…”

“…In this paper we need more precise estimates as in Ricci flow. The following result is taken from [7], and the readers are referred to [7] for details. Lemma 3.7 (cf.…”

“…Lemma 3.7 (cf. Theorem 3.2 in [7]) Assume that the Lagrangian mean curvature flow has a smooth solution for t ∈ [0, t 0 ]. If there is a constant such that…”

Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

“…In [8] one of the main tools in proving this characterization was Perelman's pseudolocality theorem [16,Theorem 10.3]. In [4] the pseudolocality theorem for the mean curvature has been proved which motivated us to prove Theorem 1.5, that is, the following: In the case of mean convex mean curvature flow, we have a stronger result, that is Theorem 1.6. The proof of this theorem is simple so we give it here first.…”

“…Therefore, Σ δ H = Σ. Before we start proving Theorem 3.1, we recall the definition of local δ-Lipschitz graph of radius r 0 and state the pseudolocality theorem from [4]. …”