Static vacuum extensions with prescribed Bartnik boundary data near a general static vacuum metric

TL;DR

This paper introduces new conditions for solving static vacuum metrics with prescribed boundary data, confirms a conjecture for many cases, and provides new examples of intriguing static vacuum geometries.

Contribution

It defines static regularity conditions ensuring local well-posedness and verifies Bartnik's conjecture for a broad class of boundary data.

Findings

Static regularity conditions are sufficient for well-posedness.

Most hypersurfaces are static regular of type (II).

Bartnik's conjecture holds for data far from Euclidean with large ADM mass.

Abstract

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik's static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry.