Asymptotically flat extensions of CMC Bartnik data

TL;DR

This paper constructs asymptotically flat 3-manifolds with prescribed boundary data, providing new bounds on the Bartnik mass by relating it to the Hawking mass and Schwarzschild geometry.

Contribution

It introduces a method to extend CMC Bartnik data to asymptotically flat manifolds with controlled mass, linking the Bartnik mass to the Hawking mass.

Findings

Mass can be made arbitrarily close to a multiple of the Hawking mass.

The multiplicative factor approaches 1 as mean curvature H tends to 0.

Provides a new upper bound for the Bartnik mass.

Abstract

Let $g$ be a metric on the $2$-sphere $\mathbb{S}^2$ with positive Gaussian curvature and $H$ be a positive constant. Under suitable conditions on $(g, H)$, we construct smooth, asymptotically flat $3$-manifolds $M$ with non-negative scalar curvature, with outer-minimizing boundary isometric to $(\mathbb{S}^2, g)$ and having mean curvature $H$, such that near infinity $M$ is isometric to a spatial Schwarzschild manifold whose mass $m$ can be made arbitrarily close to a constant multiple of the Hawking mass of $(\mathbb{S}^2,g,H)$. Moreover, this constant multiplicative factor depends only on $(g, H)$ and tends to $1$ as $H$ tends to $0$. The result provides a new upper bound of the Bartnik mass associated to such boundary data.