Vol. 292, No. 2, 2018

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ISSN: 0030-8730
Rigidity of Hawking mass for surfaces in three manifolds

Jiacheng Sun

Vol. 292 (2018), No. 2, 479–504
Abstract

It is well known that Hawking mass is nonnegative for a stable sphere of constant mean curvature (CMC) in a three-manifold of nonnegative scalar curvature. R. Bartnik proposed the rigidity problem of the Hawking mass of stable CMC spheres. We show partial rigidity results of Hawking mass for stable CMC spheres in asymptotic flat (AF) manifolds with nonnegative scalar curvature. If the Hawking mass of a nearly round stable CMC surface vanishes, then the surface must be the standard sphere in 3 and the interior of the surface is flat. Similar results also hold for asymptotic hyperbolic manifolds. A complete AF manifold having small or large isoperimetric surface with zero Hawking mass must be flat. We use the mean field equation and monotonicity of Hawking mass as well as rigidity results of Y. Shi in our proof.

Keywords
rigidity, Hawking mass, stable CMC, mean field equation
Mathematical Subject Classification 2010
Primary: 53C21, 53C24, 53C80, 58C40
Milestones
Received: 13 April 2017
Revised: 31 May 2017
Accepted: 4 June 2017
Published: 17 October 2017
Authors
Jiacheng Sun
Beijing International Center for Mathematical Research
Peking University
Quanzhai
Beijing, 100871
China