Examples of compact embedded mean convex $\lambda$-hypersurfaces

TL;DR

This paper constructs examples of compact, embedded, mean convex $\lambda$-hypersurfaces diffeomorphic to spheres for $\lambda>0$, demonstrating they are not necessarily round spheres, contrasting with the $\lambda<0$ case.

Contribution

It provides the first known examples of non-round, compact, embedded, mean convex $\lambda$-hypersurfaces for positive $\lambda$, expanding understanding of their geometric properties.

Findings

Constructed non-round, compact, embedded, mean convex $\lambda$-hypersurfaces for $\lambda>0$.

Showed that for $\lambda>0$, the only convex embedded $\lambda$-hypersphere is the round sphere.

Contrasted the $\lambda>0$ case with the $\lambda<0$ case where such examples do not exist.

Abstract

There is a well-known conjecture asserts that the round sphere should be the only compact embedded self-shrinker (i.e. $0$-hypersurface) which is diffeomorphic to a sphere. S. Brendle confirmed the conjecture for 2-dimensional $0$-hypersurfaces. For any dimensional $\lambda$-hypersurfaces, if $\lambda<0$, we constructed compact convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $\lambda>0$, we construct a compact mean convex embedded $\lambda$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $\lambda>0$, there are no compact convex embedded $\lambda$-hypersurfaces which are diffeomorphic to spheres except a round sphere.