Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

TL;DR

This paper investigates the spectral properties of self-expanding solutions to mean curvature flows, establishing bounds and uniqueness results that characterize Euclidean subspaces and hyperplanes as special self-expanders.

Contribution

It provides new spectral bounds for the drifted Laplacian on self-expanders and characterizes the Euclidean subspace and hyperplanes as unique solutions under certain conditions.

Findings

Discreteness of the spectrum of the drifted Laplacian on self-expanders.

Universal lower bound of the spectrum achieved only by Euclidean subspaces.

Uniqueness of hyperplanes through the origin for certain self-expander conditions.

Abstract

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is the Euclidian subspace through the origin. Further, for self-expanders of codimension $1$, we prove an inequality between the bottom of the spectrum of the drifted Laplacian and the bottom of the spectrum of weighted stability operator and that the hyperplane through the origin is the unique self-expander where the equality holds. Also we prove the uniqueness of hyperplane through the origin for mean convex self-expanders under some condition on the square of the norm of the second fundamental form.