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arXiv · 2603.29371

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

Qing-Ming Cheng·Junqi Lai·Guoxin Wei
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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.

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Qing-Ming Cheng, Junqi Lai, Guoxin Wei. 2026-03-31. Examples of compact embedded mean convex $\lambda$-hypersurfaces. https://arxiv.org/abs/2603.29371

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