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Upper bounds on Renormalized Volume for Schottky groups

Franco Vargas Pallete

TL;DR

The paper addresses bounding the renormalized volume $V_R$ for Schottky hyperbolic 3-manifolds with conformal boundary $\Sigma$, comparing these volumes to those of Fuchsian fillings. It develops a strategy that relates $V_R$ to the convex core volume via $V_R(M) \le V_C(M) - \tfrac{1}{4}L(\mu)$ and then bounds $V_C$ using isoperimetric inequalities and the Thurston metric to connect geometric data to extremal lengths. The key contributions are explicit upper bounds $V_R(M) \le L(\Sigma,\Gamma)^2 + \pi(g-1)$ and a sharper bound under small extremal length, which partially answer Maldacena's question by linking $V_R$ to conformal boundary data through $EL$-based quantities. This provides a concrete bridge between hyperbolic geometry, extremal length theory, and holographic-inspired gravitational action, with implications for how Schottky and Fuchsian fillings compare. The results hinge on compressing curves and optimizing over multicurves to minimize $L(\Sigma,\Gamma)$, highlighting the role of extremal length in controlling renormalized volume.

Abstract

In this article we show that for any given Riemann surface $Σ$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $Σ$ in terms of the genus and the combined extremal lengths on $Σ$ of $(g-1)$ disjoint, non-homotopic, simple closed compressible curves. This result is used to partially answer a question posed by Maldacena about comparing renormalized volumes of Schottky and Fuchsian manifolds with the same conformal boundary.

Upper bounds on Renormalized Volume for Schottky groups

TL;DR

The paper addresses bounding the renormalized volume for Schottky hyperbolic 3-manifolds with conformal boundary , comparing these volumes to those of Fuchsian fillings. It develops a strategy that relates to the convex core volume via and then bounds using isoperimetric inequalities and the Thurston metric to connect geometric data to extremal lengths. The key contributions are explicit upper bounds and a sharper bound under small extremal length, which partially answer Maldacena's question by linking to conformal boundary data through -based quantities. This provides a concrete bridge between hyperbolic geometry, extremal length theory, and holographic-inspired gravitational action, with implications for how Schottky and Fuchsian fillings compare. The results hinge on compressing curves and optimizing over multicurves to minimize , highlighting the role of extremal length in controlling renormalized volume.

Abstract

In this article we show that for any given Riemann surface of genus , we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to in terms of the genus and the combined extremal lengths on of disjoint, non-homotopic, simple closed compressible curves. This result is used to partially answer a question posed by Maldacena about comparing renormalized volumes of Schottky and Fuchsian manifolds with the same conformal boundary.

Paper Structure

This paper contains 4 sections, 5 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminary results
  4. Bounds on ${\rm V}_{\rm R}$

Key Result

Theorem 2.1

Let $\Sigma$ be a Riemann surface of genus $g$, and let $\Gamma$ be a set of $g-1$ mutually disjoint, non-homotopic, simple closed curves of $\Sigma$ with sum of square roots of extremal lengths denoted by $L(\Sigma,\Gamma)$. Moreover, assume that each component of $\Sigma\setminus\Gamma$ has genus Moreover, if we further assume that $L(\Sigma,\Gamma) \leq \sqrt{\pi(g-1)}$ then we have that whic

Figures (5)

  • Figure 1: Polygonal disk with boundary $\gamma$
  • Figure 2: Handlebody for genus $g=2$
  • Figure 3: $X\subset\mathbb{H}^3$ for genus $g=2$
  • Figure 4: $X_1$ when following adjacency only through $D^\pm_g$ for $g=2$
  • Figure 5: $Y_1$ for genus $g=2$

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 3.1: Theorem 1.1 Schlenker13, Proof of Theorem 1.2 BC15
  • Theorem 3.2: Hyperbolic isoperimetric inequality
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.1
  • ...and 1 more