Convergence of the gradient flow of renormalized volume to convex cores with totally geodesic boundary

2021/05/03 by Martin Bridgeman, Bridgeman, Martin, Kenneth Bromberg +3 · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2105.01207

openalex publication_date 2021/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures on (relatively) acylindrical manifolds. In this paper we prove the conjecture that the flow has a global attracting fixed point at the structure M_{\rm geod} the unique structure with minimum convex core volume.

Cited by

• Volume functions and boundary data of 3-dimensional hyperbolic manifolds
• Adapted Renormalized Volume for Hyperbolic 3-Manifolds with Compressible Boundary

Related

• Thom's gradient conjecture for parabolic systems and the Yang-Mills and Ricci flows
• Curvature flow of complete convex hypersurfaces in hyperbolic space
• L¹-gradient flow of convex functionals
• Mean curvature flow of mean convex hypersurfaces
• Curvature flow of complete hypersurfaces in hyperbolic space