Uniqueness and pseudolocality theorems of the mean curvature flow

By Binglong Chen, Le Yin. 2007

104 citations recorded by the source corpus.

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/96dc2d4cd2ccdb470505b5e2e1cb854332a75adb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Knut Smoczyk
2. F. Schulze
3. N. Šešum
4. A. Neves
5. Nam Q. Le
6. Huaihuo Cao
7. T. Ilmanen
8. S. Brendle

In Geometry and Topology

1. Knut Smoczyk
2. F. Schulze
3. N. Šešum
4. Nam Q. Le
5. Huaihuo Cao
6. T. Ilmanen
7. S. Brendle
8. A. Neves

Citations in the corpus

• Strong Uniqueness of the Ricci Flow (2007, methodology): “In [ 3 ], we also established the uniqueness theorem for solutions with bounded second fundamental form, hence a general strong uniqueness theorem for mean curvature flow followed as a corollary.”
• On short time existence for the planar network flow (2014, background): “A similar estimate assuming initial control on the second fundamental form has been shown by Chen and Yin [5] and assuming control on up to fourth derivatives by Brendle and Huisken [4].”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “Recently Chen and Yin [14] proved that uniqueness for complete manifolds M still holds within the class of smooth solutions with bounded second fundamental tensor, if the ambient Riemannian manifold .”
• Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers (2010, methodology): “1, we recall the definition of local δ-Lipschitz graph of radius r0 and state the pseudolocality theorem from [4].”