Rigidity of entire self-shrinking solutions to curvature flows
type: Item
instance of: scholarly article
title: Rigidity of entire self-shrinking solutions to curvature flows
zbMATH Open document ID: 1250.53064
DOI: 10.1515/CRELLE.2011.102
author: Albert Chau
author: Jingyi Chen
author: Yu Yuan
publication date: 2012-04-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1003.3246
Mathematics Subject Classification ID: 53C44
zbMATH DE Number: 6022599
zbMATH Keywords: Lagrangian mean curvature flow
zbMATH Keywords: Lagrangian graph solution
zbMATH Keywords: elliptic equation
MaRDI profile type: Publication
arXiv ID: 1003.3246
cites work: Mean curvature evolution of entire graphs
cites work: Calibrated geometries
cites work: Longtime existence of the Lagrangian mean curvature flow
published in: Journal für die Reine und Angewandte Mathematik
review text: Self-shrinking solutions of the mean curvature flow arise as ancient families of immersions from a Riemannian manifold \(\Sigma\) into \(\mathbb R^N\), \(F(x,t): \Sigma \times (-\infty, 0) \rightarrow \mathbb R^{N}\), that evolve only by scaling. In that case \(F(x,-1)\) satisfies the equation NEWLINE\[NEWLINE H+\frac{1}{2} F^\perp = 0, NEWLINE\]NEWLINE where \(H\) is the mean curvature of \(F(\Sigma,t)\) and \(\perp\) is the normal component. A Lagrangian graph solution is one of the form \(\{ (x,Du(x)): x \in \mathbb R^n \}\). The authors prove that any entire graphic self-shrinking solution of the Lagrangian mean curvature flow in \(\mathbb C^{n}\) with the Euclidean metric is flat. They then equip \(\mathbb R^{2n}\) with the indefinite metric \(\sum_{i=1}^n dx_idy_i,\) in which case the potential \(u\) of a space-like graphic self-shrinking solution is convex and satisfies an elliptic equation. They prove that if \(u\) is either radially symmetric or if its Hessian is bounded below quadratically, then the solution is flat. They further prove a Hermitian analog of this result for the Kähler Ricci flow.
reviewed by: Christine Guenther
description: scientific article; zbMATH DE number 6022599