## Abstract and Applied Analysis

### Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space

#### Abstract

Let $f(x)$ be a smooth strictly convex solution of $\text{det}({\partial }^{2}f/\partial {x}_{i}\partial {x}_{j})=\text{exp}\{(1/2){\sum }_{i=1}^{n}{x}_{i}(\partial f/\partial {x}_{i})-f\}$ defined on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^{n}$; then the graph ${M}_{\nabla f}$ of $\nabla f$ is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space ${\mathbb{R}}_{n}^{\mathrm{2}n}$ 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 ${R}_{n}^{\mathrm{2}n}$ and passes through the origin then it is flat.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 196751, 9 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607378

Digital Object Identifier
doi:10.1155/2014/196751

Mathematical Reviews number (MathSciNet)
MR3232825

Zentralblatt MATH identifier
07021914

#### Citation

Xu, Ruiwei; Cao, Linfen. Complete Self-Shrinking Solutions for Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 196751, 9 pages. doi:10.1155/2014/196751. https://projecteuclid.org/euclid.aaa/1412607378

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