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Universal Liouville action as a renormalized volume and its gradient flow

Martin Bridgeman, Kenneth Bromberg, Franco Vargas Pallete, Yilin Wang

TL;DR

The paper establishes a precise holographic link between the universal Liouville action on the Weil–Petersson Teichmüller space and the renormalized volume of the hyperbolic 3-manifold bounded by Epstein–Poincaré surfaces associated with a quasicircle. It proves the key identity $\tilde{\mathbf S}[4](\gamma)=4 V_R(\gamma)$ for sufficiently regular curves and shows $\tilde{\mathbf S}[4](\gamma) \ge 4 V_R(\gamma)$ for general Weil–Petersson quasicircles, using a Schläfli-type variational formula and an approximation by equipotentials. The gradient flow of the universal Liouville action on the Weil–Petersson component $T_0(1)$ exists for all time and converges to the round circle, yielding a quantitative bound on the Weil–Petersson distance in terms of the action. The work also clarifies the geometric positioning of Epstein–Poincaré surfaces relative to minimal surfaces and the convex hull, and it extends the variational framework to non-immersed cases, providing a robust variational theory for renormalized volumes in the universal setting.

Abstract

The universal Liouville action (also known as the Loewner energy for Jordan curves) is a Kähler potential on the Weil-Petersson universal Teichmüller space, which is identified with the family of Weil-Petersson quasicircles via conformal welding. Our main result shows that, under regularity assumptions, the universal Liouville action equals the renormalized volume of the hyperbolic $3$-manifold bounded by the two Epstein-Poincaré surfaces associated with the quasicircle. We also study the gradient descent flow of the universal Liouville action for the Weil-Petersson metric and show that the flow always converges to the origin (the circle). This provides a bound of the Weil-Petersson distance to the origin by the universal Liouville action.

Universal Liouville action as a renormalized volume and its gradient flow

TL;DR

The paper establishes a precise holographic link between the universal Liouville action on the Weil–Petersson Teichmüller space and the renormalized volume of the hyperbolic 3-manifold bounded by Epstein–Poincaré surfaces associated with a quasicircle. It proves the key identity for sufficiently regular curves and shows for general Weil–Petersson quasicircles, using a Schläfli-type variational formula and an approximation by equipotentials. The gradient flow of the universal Liouville action on the Weil–Petersson component exists for all time and converges to the round circle, yielding a quantitative bound on the Weil–Petersson distance in terms of the action. The work also clarifies the geometric positioning of Epstein–Poincaré surfaces relative to minimal surfaces and the convex hull, and it extends the variational framework to non-immersed cases, providing a robust variational theory for renormalized volumes in the universal setting.

Abstract

The universal Liouville action (also known as the Loewner energy for Jordan curves) is a Kähler potential on the Weil-Petersson universal Teichmüller space, which is identified with the family of Weil-Petersson quasicircles via conformal welding. Our main result shows that, under regularity assumptions, the universal Liouville action equals the renormalized volume of the hyperbolic -manifold bounded by the two Epstein-Poincaré surfaces associated with the quasicircle. We also study the gradient descent flow of the universal Liouville action for the Weil-Petersson metric and show that the flow always converges to the origin (the circle). This provides a bound of the Weil-Petersson distance to the origin by the universal Liouville action.
Paper Structure (22 sections, 48 theorems, 176 equations, 1 figure)

This paper contains 22 sections, 48 theorems, 176 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Universal Weil--Petersson Teichmüller space
  3. Universal Teichmüller space
  4. Kähler Structure and Weil--Petersson Teichmüller space
  5. Universal Liouville action
  6. Preliminaries on Epstein surfaces
  7. Definition of general Epstein surfaces
  8. Epstein--Poincaré surface
  9. Explicit expression of Epstein maps in the upper-space model
  10. Renormalized volume for a Jordan curve
  11. Disjoint Epstein--Poincaré surfaces
  12. Volume between the Epstein--Poincaré surfaces
  13. Volume for smooth Jordan curves
  14. Universal Liouville action as renormalized volume
  15. Schläfli formula for the variation of the volume
  16. ...and 7 more sections

Key Result

Proposition 1.1

If $\gamma$ is not a circle, then the two Epstein--Poincaré surfaces $\Sigma_{\Omega}$ and $\Sigma_{\Omega^*}$ are disjoint.

Figures (1)

  • Figure 1: Illustration of the two Epstein--Poincaré surfaces associated with the two connected component of $\hat{\mathbb{C}} \smallsetminus \gamma_t$ and the map $h_t$.

Theorems & Definitions (92)

  • Proposition 1.1: See Proposition \ref{['prop:disjoint']}
  • Proposition 1.2: See Corollary \ref{['cor:embedding_wp']}
  • Theorem 1.3
  • Theorem 1.4: See Corollary \ref{['cor:WP_HdA']}
  • Definition 1.5
  • Theorem 1.6: See Corollary \ref{['cor:smooth_identity']} and Theorem \ref{['thm:general_ineq']}
  • Theorem 1.7: See Theorem \ref{['thm:gradient']}
  • Theorem 1.8: See Theorem \ref{['thm:bound_distance']}
  • Theorem 2.1: TT06
  • Remark 2.2
  • ...and 82 more