Almost-Einstein hypersurfaces in the Euclidean space

Abstract

We show that almost-Einstein hypersurfaces in the Euclidean space are homeomorphic to spheres. The proof relies on universal lower bounds in terms of the Betti numbers for the $L^{n/2}$-norms of the Ricci and traceless Ricci tensor of compact oriented $n$-dimensional hypersurfaces. Certain examples show that the assumption on the codimension is essential.