Existence of static axisymmetric solutions to the Einstein equations
Abstract
We construct smooth Riemannian metrics and scalar potentials that are invariant under a circle action and satisfy the static Einstein equations with positive cosmological constant. The underlying compact three-manifold is a ball with two disjoint open balls removed. The potential (or lapse function) is positive in the interior and vanishes at the boundary, consisting of three non-degenerate horizons. The areas of the two inner boundary components are controlled by two arbitrary parameters, namely the masses of the black holes, which are taken to be sufficiently small and comparable. Each inner horizon radius scales linearly with the individual mass parameters, while the separation of their reference centers scales as the cube root of the total mass. These two scales are assumed to be sufficiently small at different rates, whereas the cosmological length remains fixed. The main analytical challenge is to control the linearized equation and the nonlinear remainder simultaneously near the shrinking horizons, between the centers and in the outer region. Weighted Hölder estimates quantify the growth of the inverse and show that the error of an approximation joining Kottler collars to de Sitter space is small enough to close a contraction argument. Namely, we solve a reduced elliptic Einstein equation that is endowed with two obstruction tensors; varying the two axial positions removes their coefficients. The contracted Bianchi identity and an energy estimate for an auxiliary vector field (Killing field) introduced by the projection to vanish. Finally, for a fixed cosmological constant, the intrinsic metric of one of the two boundary components is sufficient to determine a static solution with simply connected interior and non-vanishing boundary gradient, up to isometry and multiplication of the potential by a positive constant.
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BibTeX
@misc{lefloch2026existence,
title = {{Existence of static axisymmetric solutions to the Einstein equations}},
author = {Philippe G. LeFloch and Luc Rozoy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34555},
eprint = {2610.34555},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34555},
}
RIS
TY - UNPB
TI - Existence of static axisymmetric solutions to the Einstein equations
AU - Philippe G. LeFloch
AU - Luc Rozoy
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34555
UR - https://arxiv.org/abs/2610.34555
AB - We construct smooth Riemannian metrics and scalar potentials that are invariant under a circle action and satisfy the static Einstein equations with positive cosmological constant. The underlying compact three-manifold is a ball with two disjoint open balls removed. The potential (or lapse function) is positive in the interior and vanishes at the boundary, consisting of three non-degenerate horizons. The areas of the two inner boundary components are controlled by two arbitrary parameters, namely the masses of the black holes, which are taken to be sufficiently small and comparable. Each inner horizon radius scales linearly with the individual mass parameters, while the separation of their reference centers scales as the cube root of the total mass. These two scales are assumed to be sufficiently small at different rates, whereas the cosmological length remains fixed. The main analytical challenge is to control the linearized equation and the nonlinear remainder simultaneously near the shrinking horizons, between the centers and in the outer region. Weighted Hölder estimates quantify the growth of the inverse and show that the error of an approximation joining Kottler collars to de Sitter space is small enough to close a contraction argument. Namely, we solve a reduced elliptic Einstein equation that is endowed with two obstruction tensors; varying the two axial positions removes their coefficients. The contracted Bianchi identity and an energy estimate for an auxiliary vector field (Killing field) introduced by the projection to vanish. Finally, for a fixed cosmological constant, the intrinsic metric of one of the two boundary components is sufficient to determine a static solution with simply connected interior and non-vanishing boundary gradient, up to isometry and multiplication of the potential by a positive constant.
ER -