Examples of compact embedded mean convex $\hat{I}»$-hypersurfaces

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 $\hat{I}»$-hypersurfaces, if $\hat{I}»<0$, we constructed compact convex embedded $\hat{I}»$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In this paper, for $\hat{I}»>0$, we construct a compact mean convex embedded $\hat{I}»$-hypersurface which is diffeomorphic to a sphere and is not a round sphere. In fact, for $\hat{I}»>0$, there are no compact convex embedded $\hat{I}»$-hypersurfaces which are diffeomorphic to spheres except a round sphere.