A rigidity theorem of $\xi$-submanifolds in $\mathbb{C}^{2}$

TL;DR

This paper introduces $\xi$-submanifolds as a generalization of self-shrinkers and $\xi$-hypersurfaces, and proves a rigidity theorem for Lagrangian $\xi$-submanifolds in $bc^2$, advancing understanding in geometric flow theory.

Contribution

It defines $\xi$-submanifolds and establishes a new rigidity theorem for Lagrangian $\xi$-submanifolds in complex 2-space, extending previous concepts in geometric analysis.

Findings

Introduction of $\xi$-submanifolds as a generalization.

Proof of a rigidity theorem for Lagrangian $\xi$-submanifolds.

Extension of concepts from hypersurfaces to higher codimension.

Abstract

In this paper, we first introduce the concept of $\xi $-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $\lambda$-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian $\xi $-submanifold in the complex $2$-plane $\bbc^2$.