Asymptotically hyperbolic extensions and an analogue of the Bartnik mass

TL;DR

This paper develops a hyperbolic analogue of the Bartnik mass, constructing asymptotically hyperbolic extensions of certain geometric data and establishing bounds related to the hyperbolic Penrose inequality.

Contribution

It introduces a new hyperbolic version of the Bartnik mass and constructs extensions with controlled mass, extending previous flat-space results to hyperbolic settings.

Findings

Bounded the hyperbolic Bartnik mass for stable minimal surfaces.

Established upper bounds related to the hyperbolic Penrose inequality.

Provided estimates for the hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.

Abstract

The Bartnik mass is a quasi-local mass tailored to asymptotically flat Riemannian manifolds with non-negative scalar curvature. From the perspective of general relativity, these model time-symmetric domains obeying the dominant energy condition without a cosmological constant. There is a natural analogue of the Bartnik mass for asymptotically hyperbolic Riemannian manifolds with a negative lower bound on scalar curvature which model time-symmetric domains obeying the dominant energy condition in the presence of a negative cosmological constant. Following the ideas of Mantoulidis and Schoen [2016], of Miao and Xie [2016], and of joint work of Miao and the authors [2017], we construct asymptotically hyperbolic extensions of minimal and constant mean curvature (CMC) Bartnik data while controlling the total mass of the extensions. We establish that for minimal surfaces satisfying a stability condition, the Bartnik mass is bounded above by the conjectured lower bound coming from the asymptotically hyperbolic Riemannian Penrose inequality. We also obtain estimates for such a hyperbolic Bartnik mass of CMC surfaces with positive Gaussian curvature.