ALMOST-EINSTEIN HYPERSURFACES IN THE EUCLIDEAN SPACE

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.