On the Bartnik mass of CMC Bartnik data

We study the Bartnik mass in asymptotically flat Riemannian 3-manifolds with inner boundary and non-negative scalar curvature. The Bartnik mass is an important notion of "local mass" of the inner boundary, although it is notoriously difficult to compute. The problem of computing it can be rephrased as an extension problem for Bartnik data, i.e., Riemannian 2-surfaces with mean curvature H.

Recently, C. Mantoulidis and R. Schoen constructed asymptotically flat extensions of Bartnik data with H=0 allowing them to compute their Bartnik mass. We will describe how to adapt their ideas to construct extensions and obtain estimates for the Bartnik mass of Bartnik data with H a positive constant. In addition, we will discuss a Bartnik mass analog in the context of asymptotically hyperbolic manifolds, construct extensions and prove the corresponding estimates. This talk is based on joint projects with C. Cederbaum, S. McCormick, and P. Miao.