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

2022/10/04 by Jiang, Ruojing
#53A10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.01333

Abstract

On closed hyperbolic manifolds of dimension n≥ 3, we review the definition of the average area ratio of a metric h with R_h≥ -n(n-1) relative to the hyperbolic metric h₀, and we prove that it attains the local minimum of one at h₀, 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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