Bartnik's mass and Hamilton's Modified Ricci Flow

TL;DR

This paper estimates the Bartnik mass of certain spherical CMC surfaces, bounding it by the Hawking mass and a new asphericity mass derived via Hamilton's modified Ricci flow, advancing understanding of quasi-local mass in geometric analysis.

Contribution

It introduces the asphericity mass based on Hamilton's modified Ricci flow and establishes bounds for the Bartnik mass of CMC surfaces, with refined results after corrections.

Findings

Bartnik mass is bounded above by Hawking mass and asphericity mass.

The asphericity mass depends only on the surface metric, not mean curvature.

The paper studies asymptotically flat manifolds foliated by Hamilton's modified Ricci flow.

Abstract

We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton's modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton's modified Ricci flow with prescribed scalar curvature. Such manifolds were first constructed by the first author in her dissertation conducted under the supervision of M.T. Wang. We make a further study of this class of manifolds bounding the ADM masses of such manifolds and analyzing the rigid case when the Hawking mass of the inner surface of the manifold agrees with its ADM mass. New in 2020: After this paper was published, Hyun-Chul Jang observed that we dropped a term in our calculations. Tracking the consequences throughout, we see that we need only slightly change the definition of the asphericity mass and then all statements of our theorems, propositions, and lemmas remain the same as the published version with slight revisions to the proofs. Pengzi Miao observed we needed an assumption on Gauss curvature in Theorem 1. We include these corrections and also some clarifications where they are needed in blue. Both Hyun-Chul Jang and Pengzi Miao have approved of our corrections and we have sent an erratum to the journal.