Explicit rigidity of almost-umbilical hypersurfaces

Abstract

We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the traceless second fundamental form, whenever the latter is sufficiently small. Furthermore we use the inverse mean curvature flow in the hyperbolic space to deduce the best possible order of decay in the class of $C^{\infty }$-bounded hypersurfaces of the Euclidean space.

13 pages. A crucial typo in the main estimate of the first version was fixed. We added a generalisation to spaceforms as well as an optimality result. In the fourth version we corrected some minor errors and added more explanation to the main proof. Comments or discussions are very welcome

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