Convex solutions to the mean curvature flow

By Xu-jia Wang. 2004

263 citations recorded by the source corpus.

In this paper we study the classication of ancient convex solutions to the mean curvature ow in R n+1 . An open problem related to the classication of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 k n, namely whether its level set is a sphere or cylinder S k 1 R n k . In this paper we give an afrmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space R n+1 is a shrinking sphere or cylinder.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8895c346da4a9110a1f908c07b65290114baa1fc

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

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.”
Ancient asymptotically cylindrical flows and applications (2019, background): “In [Wan11], Wang proved that (up to rigid motions) the bowl soliton is the unique convex translator in R3 that is an entire graph.”
Embedded self-similar shrinkers of genus 0 (2014, background): “Wang [24] has proved a uniqueness result for convex translating solutions to the mean curvature flow which can be expressed as graphs over R3.”
Mean Curvature Flow of Mean Convex Hypersurfaces (2013, background): “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.”
Uniqueness of two-convex closed ancient solutions to the mean curvature flow (2018, methodology): “By Xu-Jia Wang’s result [17], it follows that in this case the backward limit as τ → −∞ of the type-I rescaling M̄τ of the original solution Mt, defined by (1.”
Unique asymptotics of ancient convex mean curvature flow solutions (2015, unknown): “3 in [20] implies that the dilations {X̄ ∈ R | (−t)X̄ ∈ Mt}, of hypersurfaces Mt which evolve by (1.”
Mean curvature flow with surgery of mean convex surfaces in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb (2013, methodology): “Moreover, Wang [20] has obtained a classification of translating solutions to the mean curvature flow in dimension 2.”
Ancient solutions of the mean curvature flow (2013, background): “593 of ancient flows of compact, convex sets that are not selfsimilar [24, p. 134].1 Using a related construction, phrased in the language of PDE estimates for the level set equation, Wang proved the existence of ancient convex solutions of the mean curvature flow that are not k-rotationally symmetr”
Complete translating solitons to the mean curvature flow in ℝ3 with nonnegative mean curvature (2020, methodology): “1 of [33], we may conclude COROLLARY 1.”
Translating solitons in Riemannian products (2018, background): “Wang [22] characterized the bowl soliton as the only convex translating soliton in R which is an entire graph.”