Quasihyperbolic domains are CAT(2)
Abstract
We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius π/√2 are 𝒞1,1/2-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius arctan(√2)/√2. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of −1 and is equivalent to the concavity of the domain.
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BibTeX
@misc{ikonen2026quasihyperbolic,
title = {{Quasihyperbolic domains are CAT(2)}},
author = {Toni Ikonen and Abhishek Pandey},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24610},
eprint = {2609.24610},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24610},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quasihyperbolic domains are CAT(2)
AU - Toni Ikonen
AU - Abhishek Pandey
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24610
UR - https://arxiv.org/abs/2609.24610
AB - We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius π/√2 are 𝒞^1,1/2-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius arctan(√2)/√2. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of −1 and is equivalent to the concavity of the domain.
ER -