Singularity Profile in the Mean Curvature Flow

By Weimin Sheng, Xu-jia Wang. 2009

92 citations recorded by the source corpus.

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space R n+1 with positive mean curvature is κ-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3eaa6d1c439a4af34515ef1f5b14607985d090e1

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Noncollapsing in mean-convex mean curvature flow (2012, background): “(2) @Z @xi D dHxhw; @ x i iC d(2) 2 riHxC ıdh x iqg qp x hw; @ x p i: Now the second derivatives: @(2)Z @yi@yj DHxh@ y i ; @ y j i dHxh y ij hw; yi ıh y ij h y ; xi; (3) @(2)Z @yi@xj D Hxh@ x j ; @ y i iC dhw; @ y i irj HxC ıh x jqg qp x h@ y i ; @ x p i; (4) @(2)Z @xi@xj DHxh@ x j ; @ x i i dhw; @ ”
Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature (2020, unknown): “for mean convex mean curvature flow [20, 30, 35, 36].”
Uniqueness of convex ancient solutions to mean curvature flow in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\m (2018, methodology): “A related uniqueness result for the bowl soliton was proved in an important paper by Wang [24]; this relies on a completely different approach.”
Uniqueness of the bowl soliton (2014, background): “A very important contribution was made by X. Wang [Wan11], who proved that for n = 2 any entire convex solution must be the rotationally symmetric bowl soliton, but that for n ≥ 3 there exist entire strictly convex solution that are not rotationally symmetric.”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, methodology): “In a beautiful recent paper [2], inspired by the earlier work of White [36] and Sheng and Wang [34], Andrews has given a direct proof of noncollapsing for mean convex flows using only the maximum principle.”
Embedded minimal tori in S3 and the Lawson conjecture (2012, unknown): “By applying the maximum principle to a suitable function Wt(x, y) defined on M ×M , Andrews obtained a new proof of the noncollapsing estimate established earlier by Sheng and Wang [21] (see also [25]).”
EMBEDDED CONSTANT MEAN CURVATURE TORI IN THE THREE-SPHERE (2012, result): “The estimate was inspired by earlier work of Weimin Sheng and Xujia Wang [23] (see also [28]), who in the course of a detailed analysis of singularity profiles in mean curvature flow proved that for any embedded compact meanconvex hypersurface M moving under the mean curvature flow, there is a posit”
Ancient solutions of the mean curvature flow (2013, unknown): “H ≥ 0) mean curvature flows satisfying the conclusion of Andrews’ beautiful noncollapsing result [2], see also White [23] and Sheng-Wang [21].”
Minimal surfaces in $$S^3$$: a survey of recent results (2013, background): “The noncollapsing theorem for the mean curvature flow was first stated in a paper by Sheng and Wang in [48]; the result is a direct consequence of the work of White on the structure of singularities in the mean curvature flow (cf.”