Ellipticity of the Bartnik Boundary Conditions
摘要：The Bartnik quasi-local mass is defined to measure the mass of a bounded manifold with boundary, where a collection of geometric boundary data — the so-called Bartnik boundary data— plays a key role. Bartnik proposed the open problem whether, on a given manifold with boundary, there exists a stationary vacuum metric so that the Bartnik boundary conditions are realized. In the effort to answer this question, it is important to prove the ellipticity of Bartnik boundary conditions for stationary vacuum metrics. In this talk, I will start with an introduction to the Bartnik quasi-local mass and the moduli space of stationary vacuum metrics. Then I will explain the ellipticity result for the Bartnik boundary conditions and, as an application, derive a local result to the existence question.