Stability and area growth of $\lambda$-hypersurfaces

TL;DR

This paper investigates the stability and area growth of $\lambda$-hypersurfaces by defining a new functional, extending previous results, and analyzing their geometric properties.

Contribution

It introduces an $\mathcal{F}$-functional for $\lambda$-hypersurfaces and extends stability results of Colding-Minicozzi to this class.

Findings

Established bounds on area growth for complete $\lambda$-hypersurfaces

Extended stability analysis to non-compact $\lambda$-hypersurfaces

Provided new insights into the geometric behavior of $\lambda$-hypersurfaces

Abstract

In this paper, We define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $\lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact $\lambda$-hypersurfaces are studied.