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Mathematics > Differential Geometry

arXiv:2210.01333 (math)
[Submitted on 4 Oct 2022]

Title:Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds

Authors:Ruojing Jiang
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Abstract:On closed hyperbolic manifolds of dimension $n\geq 3$, we review the definition of the average area ratio of a metric $h$ with $R_h\geq -n(n-1)$ relative to the hyperbolic metric $h_0$, and we prove that it attains the local minimum of one at $h_0$, which solves a local version of Gromov's conjecture. Additionally, we discuss the relation between the average area ratio and normalized total scalar curvature for hyperbolic $n$-manifolds, as well as its relation to the minimal surface entropy if $n$ is odd.
Comments: 28 pages. Comments are welcome!
Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS)
MSC classes: 53A10
Cite as: arXiv:2210.01333 [math.DG]
  (or arXiv:2210.01333v1 [math.DG] for this version)
  https://doi.org/10.48550/arXiv.2210.01333

Submission history

From: Ruojing Jiang [view email]
[v1] Tue, 4 Oct 2022 02:51:31 UTC (32 KB)
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