Rigidity Results for Spacelike Self-Shrinkers via Different Maximum Principles

TL;DR

This paper proves that under certain boundedness conditions, spacelike self-shrinkers in pseudo-Euclidean space must be hyperplanes, using various maximum principles to achieve classification results.

Contribution

It introduces new rigidity theorems for spacelike self-shrinkers employing different maximum principles, advancing the classification of such geometric objects.

Findings

Spacelike self-shrinkers with bounded mean curvature are hyperplanes.

Application of Omori--Yau maximum principles yields rigidity results.

Results extend classification under natural geometric conditions.

Abstract

In this work, we establish several rigidity results for spacelike self-shrinkers immersed in the pseudo-Euclidean space $\mathbb{R}^{n+p}_p$. Under suitable boundedness conditions on either the mean curvature vector or the second fundamental form, we apply different versions of Omori--Yau type maximum principles due to Qiu [20], Chen and Qiu [10], and Al\'ias, Caminha, and Nascimento [3] to show that such self-shrinkers must be spacelike hyperplanes. These results contribute to the broader classification of spacelike self-shrinkers under natural geometric assumptions.