arXiv Open Access 2023

On CAT($κ$) surfaces

Saajid Chowdhury Hechen Hu Matthew Romney Adam Tsou

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Abstrak

We study the properties of $\text{CAT}(κ)$ surfaces: length metric spaces homeomorphic to a surface having curvature bounded above in the sense of satisfying the $\text{CAT}(κ)$ condition locally. The main facts about $\text{CAT}(κ)$ surfaces seem to be largely a part of mathematical folklore, and this paper is intended to rectify the situation. We provide a complete proof that $\text{CAT}(κ)$ surfaces have bounded (integral) curvature. This fact allows one to apply the established theory of surfaces of bounded curvature to derive further properties of $\text{CAT}(κ)$ surfaces. We also show that $\text{CAT}(κ)$ surfaces can be approximated by smooth Riemannian surfaces of Gaussian curvature at most $κ$. We do this by giving explicit formulas for smoothing the vertices of model polyhedral surfaces.

Topik & Kata Kunci

math.MG math.DG

Penulis (4)

Saajid Chowdhury

Hechen Hu

Matthew Romney

Adam Tsou

Format Sitasi

APA MLA BibTeX

Chowdhury, S., Hu, H., Romney, M., Tsou, A. (2023). On CAT($κ$) surfaces. https://arxiv.org/abs/2309.13533

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Informasi Jurnal

Tahun Terbit: 2023
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓