Examples of compact embedded $\lambda$-hypersurfaces

TL;DR

This paper constructs examples of compact embedded $mbda$-hypersurfaces that are sphere-like but not isometric to standard spheres, challenging the applicability of Alexandrov's theorem to these hypersurfaces.

Contribution

The paper provides explicit examples of compact embedded $mbda$-hypersurfaces that are not isometric to spheres, demonstrating limitations of classical geometric theorems.

Findings

Existence of compact embedded $mbda$-hypersurfaces diffeomorphic to spheres

Examples are not isometric to standard spheres

Challenges the general applicability of Alexandrov's theorem

Abstract

In the paper, we construct compact embedded $\lambda$-hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. Hence, one can not expect to have Alexandrov type theorem for $\lambda$-hypersurfaces.