Extrinsic radius pinching for hypersurfaces of space forms

We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M   and the L∞L∞-norm of the second fundamental form B   such that the pinching condition tanh(R)<1‖H‖∞+C (where H is the mean curvature) implies that M is diffeomorphic to an n  -dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the LpLp-norm of H  , p⩾2p⩾2, instead of the L∞L∞-norm.