Maximum capacity of Bartnik data and a generalization of static metrics
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912901904728064 |
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| author | Jauregui, Jeffrey L. |
| author_facet | Jauregui, Jeffrey L. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12383 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximum capacity of Bartnik data and a generalization of static metrics Jauregui, Jeffrey L. Differential Geometry General Relativity and Quantum Cosmology 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. |
| title | Maximum capacity of Bartnik data and a generalization of static metrics |
| topic | Differential Geometry General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2602.12383 |