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Anosov contact metrics, Dirichlet optimization and entropy

Surena Hozoori

TL;DR

The paper classifies 3-manifolds with adapted contact metrics that minimize the Dirichlet energy, establishing a dichotomy between generalized Boothby-Wang fibrations (torsion vanishing) and algebraic Anosov contact manifolds (torsion nonzero). It reveals a deep link between Dirichlet energy minimization and Reeb dynamics, showing that the energy infimum on Anosov contact flows is governed by the measure entropy of the Reeb flow, and it derives rigidity results tied to $SL(2,\mathbb{R})$-models. The authors connect variational aspects to curvature realization, proving global obstructions and Ricci-Reeb realizability formulas in the algebraic Anosov setting. Overall, the work highlights a striking interplay between geometric analysis on contact manifolds and the ergodic theory of Anosov flows, with implications for entropy rigidity and curvature problems.

Abstract

The first main result of this paper classifies contact 3-manifolds admitting critical metrics, i.e. adapted metrics which are the critical points of the Dirichlet energy functional. This gives a complete answer to a question raised by Chern-Hamilton in 1984. Secondly, we show that in the case of Anosov contact metrics, the optimization of such energy functional is closely related to Reeb dynamics and can be described in terms of its entropy. We also study the consequences in the curvature realization problem for such contact metrics.

Anosov contact metrics, Dirichlet optimization and entropy

TL;DR

The paper classifies 3-manifolds with adapted contact metrics that minimize the Dirichlet energy, establishing a dichotomy between generalized Boothby-Wang fibrations (torsion vanishing) and algebraic Anosov contact manifolds (torsion nonzero). It reveals a deep link between Dirichlet energy minimization and Reeb dynamics, showing that the energy infimum on Anosov contact flows is governed by the measure entropy of the Reeb flow, and it derives rigidity results tied to -models. The authors connect variational aspects to curvature realization, proving global obstructions and Ricci-Reeb realizability formulas in the algebraic Anosov setting. Overall, the work highlights a striking interplay between geometric analysis on contact manifolds and the ergodic theory of Anosov flows, with implications for entropy rigidity and curvature problems.

Abstract

The first main result of this paper classifies contact 3-manifolds admitting critical metrics, i.e. adapted metrics which are the critical points of the Dirichlet energy functional. This gives a complete answer to a question raised by Chern-Hamilton in 1984. Secondly, we show that in the case of Anosov contact metrics, the optimization of such energy functional is closely related to Reeb dynamics and can be described in terms of its entropy. We also study the consequences in the curvature realization problem for such contact metrics.
Paper Structure (11 sections, 30 theorems, 89 equations)

This paper contains 11 sections, 30 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Contact metrics and Dirichlet energy
  3. Compatible Riemannian geometry for contact 3-manifolds
  4. Torsion, Dirichlet functional and critical contact metrics
  5. Elements from geometry and ergodic theory of Anosov contact 3-flows
  6. Important examples
  7. Boothby-Wang fibrations
  8. Examples with $\widetilde{SL}(2,\mathbb{R})$-geometry
  9. Critical contact metrics and the Chern-Hamilton question
  10. Anosovity of contact metrics and energy optimization as asymptotic synchronization
  11. Curvature of Anosov contact metrics

Key Result

Theorem 1.2

(Theorem class) Let $(M,\alpha)$ be a contact 3-manifold such that the Dirichlet energy functional admits a minimizer. Then, (i) $(M,\alpha)$ can be approximated by generalized Boothby-Wang fibrations (the case $||\tau||\equiv 0$); or (ii) $(M,\alpha)$ is smoothly strictly contactomorphic to an alge

Theorems & Definitions (45)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Example 1.9
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 35 more