Convexity of $\lambda$-hypersurfaces

TL;DR

This paper proves that closed mean convex $\lambda$-hypersurfaces are convex when $\lambda extless= 0$, extending previous results and establishing a gap theorem for such hypersurfaces.

Contribution

It generalizes Guang's work to higher dimensions and $\lambda extless= 0$, providing new convexity and gap theorems for $\lambda$-hypersurfaces.

Findings

Closed mean convex $\lambda$-hypersurfaces are convex for $\lambda extless= 0$

Established a gap theorem for closed $\lambda$-hypersurfaces with $\lambda extless= 0$

Extended convexity results to higher dimensions and broader $\lambda$ ranges

Abstract

We prove that any $n$-dimensional closed mean convex $\lambda$-hypersurface is convex if $\lambda\le 0.$ This generalizes Guang's work on $2$-dimensional strictly mean convex $\lambda$-hypersurfaces. As a corollary, we obtain a gap theorem for closed $\lambda$-hypersurfaces with $\lambda\le 0.$