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Maximum capacity of Bartnik data and a generalization of static metrics

Jeffrey L. Jauregui

Abstract

Inspired by R. Bartnik's mass minimization problem in general relativity, we investigate a dual problem of maximizing the capacity among asymptotically flat extensions (with nonnegative scalar curvature) of some fixed two-dimensional boundary data. Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions, we derive the variational condition satisfied by a maximal capacity extension. The resulting equation is an inhomogeneous generalization of the well-known static equation, now coupled with the Baird--Eells stress-energy tensor for a harmonic function. We analyze these ``harmonic-static'' metrics in a local sense, proving they have constant scalar curvature and serve as critical points for a metric-dependent Dirichlet energy functional. We conclude with a number of open questions.

Maximum capacity of Bartnik data and a generalization of static metrics

Abstract

Inspired by R. Bartnik's mass minimization problem in general relativity, we investigate a dual problem of maximizing the capacity among asymptotically flat extensions (with nonnegative scalar curvature) of some fixed two-dimensional boundary data. Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions, we derive the variational condition satisfied by a maximal capacity extension. The resulting equation is an inhomogeneous generalization of the well-known static equation, now coupled with the Baird--Eells stress-energy tensor for a harmonic function. We analyze these ``harmonic-static'' metrics in a local sense, proving they have constant scalar curvature and serve as critical points for a metric-dependent Dirichlet energy functional. We conclude with a number of open questions.
Paper Structure (19 sections, 16 theorems, 69 equations)

This paper contains 19 sections, 16 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Finiteness results and bounds
  3. Technical setup
  4. Variational characterization of maximum capacity extensions
  5. Harmonic-static metrics: local analysis and interpretation
  6. Examples
  7. Example 1:
  8. Example 2:
  9. Example 3:
  10. Example 4:
  11. Discussion and questions
  12. Quasi-local capacity
  13. Upper semicontinuity of capacity
  14. Questions and conjectures
  15. Appendix
  16. ...and 4 more sections

Key Result

Proposition 6

Let $(\Sigma, \gamma, H)$ be Bartnik data. Then where $|\Sigma|_{\gamma}$ is the area of $\Sigma$ with respect to $\gamma$. In particular, $\mathcal{C}(\Sigma, \gamma, H) < \infty$.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5
  • Remark 2
  • Remark 3
  • Proposition 6
  • proof
  • ...and 34 more