Mathematical Research Letters

Volume 25 (2018)

Number 2

Hausdorff stability of the round two-sphere under small perturbations of the entropy

Pages: 347 – 365



Jacob Bernstein (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Lu Wang (Department of Mathematics, University of Wisconsin, Madison, Wi., U.S.A.)


We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.

The first author was partially supported by the NSF Grants DMS-1307953 and DMS-1609340. The second author was partially supported by the NSF Grant DMS-1406240 and an Alfred P. Sloan Research Fellowship. Support for the second author’s research was also provided by the Vice Chancellor for the Research and Graduation Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation. This material is based upon work supported by the NSF Grant DMS-1440140 while the second author was in residence at the Mathematical Sciences Research Institute (MSRI) in Berkeley, CA, during the Spring 2016 semester.

Received 23 August 2016

Accepted 30 April 2017

Published 5 July 2018