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

TL;DR

This paper proves that the gradient flow of renormalized volume in hyperbolic structures converges to the convex core with minimal volume, confirming a key conjecture in geometric topology.

Contribution

It establishes the global convergence of the Weil-Petersson gradient flow to the minimal convex core structure in hyperbolic manifolds.

Findings

Flow has a unique global attracting fixed point at the minimal convex core structure.

Gradient flow converges to the convex core with totally geodesic boundary.

Confirms the conjecture about the flow's convergence in hyperbolic geometry.

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.