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Adapted Renormalized Volume for Hyperbolic 3-Manifolds with Compressible Boundary

Viola Giovannini

TL;DR

This work extends the theory of renormalized volume to convex cocompact hyperbolic 3-manifolds with compressible boundary by introducing an adapted renormalized volume $ ilde{V}_R$ that remains bounded below, has controlled differential and Weil–Petersson gradient, and stays close to the convex core volume. The authors develop a pointed-volume framework to handle cusp-like boundary data and prove continuity of $ ilde{V}_R$ on strata of the WP completion corresponding to compressible multicurves, including behavior under pinchings. They provide explicit analyses of cusps and long tubes via Epstein surfaces and $W$-volumes, establish Polyakov-type variation formulas, and show that under pinching, $ ilde{V}_R$ converges to the sum of adapted volumes of pointed limit manifolds. The results yield quantitative relations between $ ilde{V}_R$ and $V_C$, yield bounds for handlebody convex cores, and offer a robust geometric-analytic tool for studying degenerations in Teichmüller and WP geometry with marked boundary data.

Abstract

The renormalized volume is a smooth function associating to every convex co-compact hyperbolic $3$-manifold $M$ a real number. When the boundary of $M$ is incompressible, the renormalized volume is always positive, otherwise there are sequences of convex co-compact structures on $M$ whose renormalized volumes diverge to minus infinity. We define here a new version of the renormalized volume which adapts to the compressible boundary case, satisfying properties analogous to those of the classical one in the incompressible setting. In particular, the adapted renormalized volume is bounded from below, its differential has uniformly bounded supremum norm, and its gradient has uniformly bounded Weil-Petersson norm. Moreover, it stays at uniformly bounded distance from the convex core volume function. As a corollary, we obtain a bound on the convex core volume of handlebodies in terms of the Weil-Petersson distance from a certain subset of the Teichmüller space, where the convex core volume is bounded by a known constant. Furthermore, the adapted renormalized volume extends continuously, as a function on the Teichmüller space of $\partial\bar M$, to the strata in the boundary of its Weil-Petersson completion corresponding to compressible multicurves. We provide a geometric interpretation of the limit quantity by defining a renormalized volume, and its adapted version, for convex co-compact hyperbolic $3$-manifolds with a finite set of marked points in the boundary.

Adapted Renormalized Volume for Hyperbolic 3-Manifolds with Compressible Boundary

TL;DR

This work extends the theory of renormalized volume to convex cocompact hyperbolic 3-manifolds with compressible boundary by introducing an adapted renormalized volume that remains bounded below, has controlled differential and Weil–Petersson gradient, and stays close to the convex core volume. The authors develop a pointed-volume framework to handle cusp-like boundary data and prove continuity of on strata of the WP completion corresponding to compressible multicurves, including behavior under pinchings. They provide explicit analyses of cusps and long tubes via Epstein surfaces and -volumes, establish Polyakov-type variation formulas, and show that under pinching, converges to the sum of adapted volumes of pointed limit manifolds. The results yield quantitative relations between and , yield bounds for handlebody convex cores, and offer a robust geometric-analytic tool for studying degenerations in Teichmüller and WP geometry with marked boundary data.

Abstract

The renormalized volume is a smooth function associating to every convex co-compact hyperbolic -manifold a real number. When the boundary of is incompressible, the renormalized volume is always positive, otherwise there are sequences of convex co-compact structures on whose renormalized volumes diverge to minus infinity. We define here a new version of the renormalized volume which adapts to the compressible boundary case, satisfying properties analogous to those of the classical one in the incompressible setting. In particular, the adapted renormalized volume is bounded from below, its differential has uniformly bounded supremum norm, and its gradient has uniformly bounded Weil-Petersson norm. Moreover, it stays at uniformly bounded distance from the convex core volume function. As a corollary, we obtain a bound on the convex core volume of handlebodies in terms of the Weil-Petersson distance from a certain subset of the Teichmüller space, where the convex core volume is bounded by a known constant. Furthermore, the adapted renormalized volume extends continuously, as a function on the Teichmüller space of , to the strata in the boundary of its Weil-Petersson completion corresponding to compressible multicurves. We provide a geometric interpretation of the limit quantity by defining a renormalized volume, and its adapted version, for convex co-compact hyperbolic -manifolds with a finite set of marked points in the boundary.

Paper Structure

This paper contains 26 sections, 38 theorems, 302 equations, 4 figures.

Table of Contents

  1. Introduction and Results
  2. Outline of the paper
  3. Preliminaries
  4. Epstein surfaces and W-volumes
  5. Epstein surfaces: construction and properties
  6. Epstein surfaces coordiantes
  7. Fundamental forms
  8. Epstein surfaces of domains with boundary
  9. W-volume and Renormalized volume
  10. W-volume for domains with round boundary
  11. Norms on quadratic differential and the Weil-Petersson pairing
  12. The boundary of the Weil-Petersson completion
  13. Conformally equivalent surfaces with marked points
  14. Thin Tubes
  15. Geometric convergence
  16. ...and 11 more sections

Key Result

Theorem 1.2

(Ahlfors, Bers, Kra, Marden, Maskit, Sullivan, Thurston, Ma2016). The deformation space $CC(M)$ of convex co-compact hyperbolic structures on $M$ considered up to homotopy is biholomorphically parameterized by a quotient of the space of conformal structures on the boundary at infinity, as where $\mathcal{T}(\partial\overline M)$ is the product of the Teichmüller spaces of the connected components

Figures (4)

  • Figure 1: Epstein surface of the infinite cusp on the punctured disk with round boundary.
  • Figure 2: The Epstein surfaces associated to the interior of $\mathbb{D}_{\rho_1}^{\rho_2}$ equipped with the hyperbolic metric $\hat{I_0}$, for two different examples of radii $\rho_i$. Note that $0.3 > e^{-\sqrt{2}} > 0.1$.
  • Figure 3: The Epstein surface associated to the domain with boundary $(\mathbb{D}_{\rho_1}^{\rho_2}, \hat{I_0})$: in yellow the Epstein surface $\Sigma_{\rho_1}^{\rho_2}(\hat{I_0})$ of the interior of the annulus, in purple the two caterpillar regions $C_i$, and in light brown the two hyperplanes $H_i$ of euclidean radii $\rho_i$.
  • Figure 4: Approximating a cusp with long truncated half-tubes of core lengths $\ell_1 > \ell_2 > \ell_3$.

Theorems & Definitions (95)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 85 more