Some results on space-like self-shrinkers

TL;DR

This paper investigates space-like self-shrinkers in pseudo-Euclidean spaces, deriving geometric properties, volume estimates, and rigidity results under growth conditions, contributing to the understanding of their structure and classification.

Contribution

It introduces new formulas for drift Laplacian of geometric quantities and establishes rigidity theorems for space-like self-shrinkers in pseudo-Euclidean spaces.

Findings

Derived drift Laplacian formulas for geometric quantities.

Established volume estimates in pseudo-distance functions.

Proved rigidity results under growth conditions of mean curvature or Gauss map images.

Abstract

We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space $\ir{m+n}_m$with index $m$. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove a rigidity results under minor growth conditions interms of the mean curvature or the image of Gauss maps.