Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Bartnik extension problem for the static vacuum Einstein
  equations   We develop a framework for understanding the existence of asymptotically flat
solutions to the static vacuum Einstein equations with prescribed boundary data
consisting of the induced metric and mean curvature on a 2-sphere. A partial
existence result is obtained, giving a partial resolution of a conjecture of
Bartnik on such static vacuum extensions. The existence and uniqueness of such
extensions is closely related to Bartnik's definition of quasi-local mass.

29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex