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

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 $\bbc^2$.