Asian Journal of Mathematics

Volume 22 (2018)

Number 6

Explicit rigidity of almost-umbilical hypersurfaces

Pages: 1075 – 1088



Julien Roth (Laboratoire d’Analyse et de Mathématiques Appliquées, UPEM-UPEC, CNRS, Marnela-Vallée, France)

Julian Scheuer (Albert-Ludwigs-Universität, Mathematisches Institut, Freiburg, Germany)


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.


pinching, almost-umbilical hypersurfaces, inverse mean curvature flow

2010 Mathematics Subject Classification

53C20, 53C21, 53C24, 58C40

Received 4 December 2015

Accepted 3 March 2017

Published 6 February 2019