Almost-Einstein hypersurfaces in the Euclidean space. (English) Zbl 1213.53072

The author shows that almost-Einstein hypersurfaces \((M^n,g), n\geq 3\) (in the sense that \(\text{Ric}_g-kg\) or \(\text{Ric}_g-\frac{s_g}{n}g\) are small in the \(L^{n/2}\)-norm for a constant \(k\geq 0\), the Ricci tensor \(\text{Ric}_g\) and the scalar curvature \(s_g\)) in the Euclidean space \(\mathbb R^{n+1}\) are homeomorphic to spheres \(S^n\). The idea for the proofs relies on universal lower bounds in terms of the Betti numbers for the \(L^{n/2}\)-norms of the Ricci and the traceless Ricci tensor of compact oriented \(n\)-dimensional hypersurfaces. Moreover, certain counterexamples show that the assumption on the codimension to be 1 is essential.


53C40 Global submanifolds
53C20 Global Riemannian geometry, including pinching
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: Euclid


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