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PreprintCC BY 4.0arXiv

Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space

  • Wei Zhang
  • Qi Zhou
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Abstract

In this paper, we study the first Dirichlet eigenfunction on bounded, smooth and strictly horo-convex domains in hyperbolic space ℍn. We prove its strict log-concavity in arbitrary dimension and establish the Gauss curvature bound K>1 for its level sets. In dimensions two and three, we further obtain quantitative lower bounds for K in terms of the minimum Gauss curvature of the boundary. In higher dimensions, the product of the two smallest principal curvatures of each level hypersurface exceeds one. Furthermore, we extend the strict log-concavity result to bounded smooth horo-convex domains. Our approach combines maximum or minimum principles for suitable auxiliary functions, boundary convexity estimates and a domain deformation argument.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026logconcavity,
  title = {{Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space}},
  author = {Wei Zhang and Qi Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07907},
  eprint = {2611.07907},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07907},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Log-concavity and level set curvature estimates for the first Dirichlet eigenfunction in hyperbolic space
AU  - Wei Zhang
AU  - Qi Zhou
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.07907
UR  - https://arxiv.org/abs/2611.07907
AB  - In this paper, we study the first Dirichlet eigenfunction on bounded, smooth and strictly horo-convex domains in hyperbolic space ℍ^n. We prove its strict log-concavity in arbitrary dimension and establish the Gauss curvature bound K>1 for its level sets. In dimensions two and three, we further obtain quantitative lower bounds for K in terms of the minimum Gauss curvature of the boundary. In higher dimensions, the product of the two smallest principal curvatures of each level hypersurface exceeds one. Furthermore, we extend the strict log-concavity result to bounded smooth horo-convex domains. Our approach combines maximum or minimum principles for suitable auxiliary functions, boundary convexity estimates and a domain deformation argument.
ER  -