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

Ruojing Jiang

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arXiv · 2210.01333

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

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.

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Ruojing Jiang. 2022-10-04. Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds. https://arxiv.org/abs/2210.01333

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