Complete self-shrinking solutions for Lagrangian mean curvature flow in pseudo-Euclidean space
type: Item
instance of: scholarly article
title: Complete self-shrinking solutions for Lagrangian mean curvature flow in pseudo-Euclidean space
zbMATH Open document ID: 1476.53114
DOI: 10.1155/2014/196751
author: Ruiwei Xu
author: Linfen Cao
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: Let \(f(x)\) be a smooth strictly convex solution of \(\text{det}(\partial^2 f / \partial x_i \partial x_j) = \text{exp} \left\{(1 / 2) \sum_{i = 1}^n x_i(\partial f / \partial x_i) - f\right\}\) defined on a domain \(\Omega \subset \mathbb{R}^n\); then the graph \(M_{\nabla f}\) of \(\nabla f\) is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space \(\mathbb{R}_n^{2 n}\) with the indefinite metric \(\sum d x_i d y_i\). In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph \(M_{\nabla f}\) is complete in \(R_n^{2 n}\) and passes through the origin then it is flat.
Mathematics Subject Classification ID: 53E10
Mathematics Subject Classification ID: 53A35
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 53E50
zbMATH DE Number: 7021914
zbMATH Keywords: space-like self-shrinker
zbMATH Keywords: pseudo-Euclidean space
zbMATH Keywords: complete self-shrinkers
zbMATH Keywords: Lagrangian graph
Wikidata QID: Q59035963
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/196751
OpenAlex ID: W2057168028
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description: scientific article; zbMATH DE number 7021914