Regularity Theory for Mean Curvature Flow

By Klaus H. Ecker. 2003

444 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b34e80c97f54215972901d5d2b18873f09d77462

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. Robert Haslhofer
2. S. Brendle
3. Lu Wang
4. Bruce Kleiner
5. F. Schulze
6. W. Minicozzi
7. T. Colding
8. A. Neves

In Geometry and Topology

1. Robert Haslhofer
2. S. Brendle
3. Bruce Kleiner
4. Lu Wang
5. W. Minicozzi
6. T. Colding
7. G. Huisken
8. Knut Smoczyk

Citations in the corpus

• Uniqueness of Self-similar Shrinkers with Asymptotically Conical Ends (2011, methodology): “17) follows from the divergence theorem and integration by parts together with applications of the Cauchy-Schwarz inequality and the basic property for backwards mean curvature flow (see Appendix B of [10])”
• A Bernstein type theorem for self-similar shrinkers (2009, result): “Similar to the arguments of Lemma 3 in [5] and [6], we construct suitable open balls as barriers.”
• 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, background): “A proof of this fact can be found in [5], pp.”
• Mean Curvature Flow in Higher Codimension: Introduction and Survey (2011, background): “A nice reference for some of the scaling techniques is [28].”
• The phase field method for geometric moving interfaces and their numerical approximations (2019, background): “The MCF is a curve-shortening (in 2-D) and an area minimization flow (in higher dimensions) in the sense that it can be interpreted as a gradient flow for the curve length functional (in 2-D) and the surface area functional (in higher dimensions) [466, 167, 230].”
• On short time existence for the planar network flow (2014, background): “We follow here the alternative proof of Ecker [7, Theorem 5.6].”
• Self-similar solutions to the curve shortening flow (2010, background): “However, in [E], it is shown that in this case the curve is the (possibly rescaled) graph of the function x 7→ − log(cosx), the so-called Grim Reaper curve; see Figure 1.”
• Uniqueness and pseudolocality theorems of the mean curvature flow (2007, background): “White [25] (see [10]) in different situations.”
• Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “1) plays a crucial role, for instance, in the derivation of the equation for flow by mean curvature, see for instance [37].”
• Embedded self-similar shrinkers of genus 0 (2014, background): “Applying Proposition 4.9 in [12] (with t 0 = 0 and ρ = 4) gives area( M t ∩ {| x | ≤ 2 } ) ≤ 8 area( M − 1 ∩ {| x | ≤ 4 } ) for all t ∈ [ − 1 , 0).”
• A general regularity theory for weak mean curvature flow (2011, methodology): “The good reference on the subject is [9], where one can find many useful estimates for Brakke’s flow written in smooth setting.”
• Mean curvature flow with surgeries of two–convex hypersurfaces (2009, unknown): “Also, an analysis of a flow through singularities of twodimensional mean convex surfaces has been announced in [5], while geometric information on the behavior of weak solutions of mean curvature flow can be found in [6, 27, 28].”
• The singular set of mean curvature flow with generic singularities (2014, background): “These results were surveyed in [17]; see also [6,20], and [35] for regularity theory in MCF.”