Some rigidity properties for $\lambda$-self-expanders

TL;DR

This paper extends known results about self-expanders to a broader class called $\lambda$-self-expanders, characterizing their geometric properties and analyzing area growth under curvature constraints.

Contribution

It generalizes previous findings on self-expanders to $\lambda$-self-expanders, including characterizations and area growth analysis.

Findings

Hyperplanes, spheres, and cylinders are characterized as $\lambda$-self-expanders.

Area growths are analyzed under mean curvature control.

Finiteness of weighted areas is discussed.

Abstract

$\lambda$-self-expanders $\Sigma$ in $\mathbb{R}^{n+1}$ are the solutions of the isoperimetric problem with respect to the same weighted area form as in the study of the self-expanders. In this paper, we mainly extend the results on self-expanders which we obtained in \cite{ancari2020volum} to $\lambda$-self-expanders. We prove some results that characterize the hyperplanes, spheres and cylinders as $\lambda$-self-expanders. We also discuss the area growths and the finiteness of the weighted areas under the control of the growth of the mean curvature.