The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores

Abstract

In this paper, we use the Weil-Petersson gradient flow for renormalized volume to study the space $CC(N;S,X)$ of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold $(N;S)$. Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasi-Fuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure $M_{\mathrm{geod}}\in CC(N;S,X)$ with totally geodesic convex core boundary facing $S$. Analyzing the geometry of structures along a flow line, we show that if $V_R(M)$ is the renormalized volume of $M$, then $V_R(M)-V_R(M_{\mathrm{geod}})$ is bounded below by a linear function of the Weil-Petersson distance $d_{\mathrm{WP}}({\partial }_{c}M,{\partial }_c{M}_{\mathrm{geod}})$, with constants depending only on the topology of $S$. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm's result that $M_{\mathrm{geod}}$ has minimal volume for $N$ acylindrical and the second author's result comparing convex core volume and Weil-Petersson distance for quasifuchsian manifolds.

References in corpus (3)

• Geometry of the Weil-Petersson completion of Teichmüller space
• On the renormalized volume of hyperbolic 3-manifolds
• Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

Cited by in corpus (1)

• Compactification and distance on Teichmüller space via renormalized volume