Let Σ be a smooth closed hypersurface with non-negative Ricci curvature, isometrically immersed in a space form. It has been proved in [P], [CZ], and [C2] that there are some L 2 inequalities on Σ which measure the stability of closed umbilical hypersurfaces or more generally, closed hypersurfaces with traceless Newton transformation of the second fundamental form. In this paper, we prove that the constants in these inequalities are optimal.
Euclidean space with the flat metric. We compute the isoperimetric profile of (T 2 × R n , h 2 + g E ), 2 ≤ n ≤ 5, for small and big values of the volume. These computations give explicit lower bounds for the isoperimetric profile of T 2 × R n . We also note that similar estimates for (We compute this explicitly for k = 3. We use symmetrization techniques for product manifolds, based on work of A. Ros ([19]) and F. Morgan ([10]).
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