Volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$
Xue Hu, Dandan Ji, Yuguang Shi
TL;DR
This paper proves a local volume comparison and rigidity results for conformally compact manifolds with scalar curvature bounds using Ricci-DeTurk flow, extending understanding of their geometric stability.
Contribution
It introduces a stability result for conformally compact Einstein manifolds and applies Ricci-DeTurk flow to establish volume comparison and rigidity theorems.
Findings
Stability of conformally compact Einstein manifolds under Ricci-DeTurk flow
Local volume comparison for manifolds with scalar curvature ≥ -n(n-1)
Rigidity when the renormalized volume is zero
Abstract
In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature and also the rigidity result when certain renormalized volume is zero.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Nonlinear Partial Differential Equations