Rigidity of closed $\lambda$-self-expanders

Abstract

A $\lambda$-self-expander $x:M^n\to {\mathbb{R}}^{n+1}$ is the solution of the isoperimetric problem of weight $e^{\frac{|x{|}^2}{4}}$. In this paper, we prove that the closed $\lambda$-self-expander is a round sphere centered at the origin under some curvature conditions. These curvature conditions mainly include the scalar curvature, the mean curvature, and the squared norm of the second fundamental form.

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