The Rigidity Theorem of Legendrian self-shrinkers

TL;DR

This paper establishes a rigidity theorem for Legendrian self-shrinkers by analyzing their volume growth, providing new insights into their geometric properties in contact Euclidean spaces.

Contribution

It introduces an optimal volume growth estimate for Legendrian self-shrinkers, leading to a rigidity result for entire smooth solutions in standard contact Euclidean spaces.

Findings

Optimal volume growth estimate for Legendrian self-shrinkers

Rigidity theorem for entire smooth Legendrian self-shrinkers

Enhanced understanding of geometric constraints in contact Euclidean spaces

Abstract

By estimating the weighted volume, we obtain the optimal volume growth for Legendrian self-shrinkers. This, in turn, yields a rigidity theorem for entire smooth Legendrian self-shrinkers in the standard contact Euclidean (2n+1)-space.