It is well known that Hawking mass is nonnegative for a stable constant mean curvature
(CMC) sphere 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
and
the interior of the surface is flat. Similar results also hold for asymptotic hyperbolic
manifolds. A complete AF manifold which has 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
