Uniqueness of the simply connected static three-dimensional hemisphere
Let $M$ be a compact, connected, simply connected smooth three-manifold with nonempty connected boundary, $g$ a smooth Riemannian metric on $M$ and $f \in C^\infty(M)$ with $$R_g = 6, \qquad \nabla_g^2 f = f(\operatorname{Ric}_g - 3g), \qquad f > 0 \text{ in } \operatorname{int} M, \qquad f^{-1}(0) = \partial M, \qquad \max_M f = 1.$$ Let $\mathbb{S}^3_+ = \{x \in \mathbb{S}^3 : x_4 \ge 0\}$ have unit sectional curvature; with $f = x_4$ it satisfies these conditions.
Ambrozio proved that a compact simply connected static triple with connected boundary and scalar curvature $6$ is the hemisphere if its boundary has area at least $4\pi/3$. He also proved that a compact static triple with scalar curvature $6$ and nonnegative Ricci curvature is the hemisphere or is covered by the cylinder $[0, \pi/\sqrt{3}] \times \mathbb{S}^2$ with metric $dt^2$ plus $\frac{1}{3}$ times the unit round metric, and he observes that a quotient of this cylinder is an orientable example with connected boundary.
Problem. Must there be an isometry $F : (M, g) \to (\mathbb{S}^3_+, g_{\mathrm{round}})$ with $f = x_4 \circ F$?
Source: Ambrozio (2017), the question following Remark 3 (numbering of arXiv:1503.03803v1), in the setting of the cosmic no-hair conjecture, suggested by Boucher, Gibbons and Horowitz (1984) and stated by Boucher and Gibbons (1983). Normalised (R = 6, max f = 1) by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, hemispheres, scalar curvature, static manifolds
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{ambrozio2017,
author = {Ambrozio, Lucas},
title = {On static three-manifolds with positive scalar curvature},
journal = {J. Differential Geom.},
volume = {107},
number = {1},
year = {2017},
pages = {1--45},
doi = {10.4310/jdg/1505268028},
eprint = {1503.03803},
archivePrefix = {arXiv},
}
@article{boucher1984,
author = {Boucher, W. and Gibbons, G. W. and Horowitz, Gary T.},
title = {Uniqueness theorem for anti-de {S}itter spacetime},
journal = {Phys. Rev. D},
volume = {30},
number = {12},
year = {1984},
pages = {2447--2451},
doi = {10.1103/PhysRevD.30.2447},
}
@misc{aim-uniqueness-of-the-simply-connected-static-three-dimensional-hemisphere,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the simply connected static three-dimensional hemisphere}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/881-uniqueness-of-the-simply-connected-static-three-dimensional-hemisphere.md},
note = {Entry 881 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01129, "Uniqueness of the simply connected static three-dimensional hemisphere", clopen.solutions.