Hyperbolic comparison and rigidity for the hyperspherical radius
Abstract
We prove Gromov's hyperbolic hyperspherical radius comparison and rigidity conjectures in all dimensions n≥3, without the spin assumption. Let (Xn, g) be a smooth, oriented, connected, compact Riemannian manifold with nonempty smooth connected boundary S. If Scg≥ −n(n−1) and HS≥ (n−1)coth R for some R>0, then Rad(S,gS)≤sinh R, with equality if and only if (Xn,g) is isometric to the closed hyperbolic geodesic ball of radius R. Under the same curvature bounds, every smooth $1$-Lipschitz map of nonzero degree from S to the boundary of the hyperbolic geodesic ball of radius R extends to an isometry of the fillings. Here, gS is the induced boundary metric, HS is the mean curvature of S with respect to the outward unit normal, and Rad(S,gS) denotes the hyperspherical radius.
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BibTeX
@misc{jiang2026hyperbolic,
title = {{Hyperbolic comparison and rigidity for the hyperspherical radius}},
author = {Xumin Jiang and Zhehui Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28450},
eprint = {2610.28450},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28450},
}
RIS
TY - UNPB
TI - Hyperbolic comparison and rigidity for the hyperspherical radius
AU - Xumin Jiang
AU - Zhehui Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28450
UR - https://arxiv.org/abs/2610.28450
AB - We prove Gromov's hyperbolic hyperspherical radius comparison and rigidity conjectures in all dimensions n≥3, without the spin assumption. Let (X^n, g) be a smooth, oriented, connected, compact Riemannian manifold with nonempty smooth connected boundary S. If Sc_g≥ −n(n−1) and H_S≥ (n−1)coth R for some R>0, then Rad(S,g_S)≤sinh R, with equality if and only if (X^n,g) is isometric to the closed hyperbolic geodesic ball of radius R. Under the same curvature bounds, every smooth $1$-Lipschitz map of nonzero degree from S to the boundary of the hyperbolic geodesic ball of radius R extends to an isometry of the fillings. Here, g_S is the induced boundary metric, H_S is the mean curvature of S with respect to the outward unit normal, and Rad(S,g_S) denotes the hyperspherical radius.
ER -