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Ergodic and Entropic Behavior of the Harmonic Map Heat Flow to the Moduli Space of Flat Tori

Mohammad Javad Habibi Vosta Kolaei

TL;DR

The paper analyzes the harmonic map heat flow from a compact manifold $M$ into the moduli space $\mathcal{M}_1 \cong \mathrm{SL}(2,\mathbb{Z}) \backslash \mathbb{H}$ of unit-area flat tori, equipped with its natural hyperbolic geometry. It shows that the flow is energy-dissipative and converges to a harmonic map, while the pushforward measures of the evolving maps converge weak-* to the hyperbolic measure $\mu_{\mathrm{hyp}}$ on $\mathcal{M}_1$; the long-time distribution of the image is statistically uniform across the moduli space. A relative-entropy framework is developed, proving $H(\mu_t|\mu_{\mathrm{hyp}}) \to 0$, which provides a quantitative refinement of ergodicity and links geometric flow dynamics with information-theoretic convergence. The work integrates harmonic map flow techniques with Teichmüller dynamics and entropy methods to connect geometric analysis, moduli-space dynamics, and probabilistic convergence in a rigorous, quantitative fashion.

Abstract

We investigate the harmonic map heat flow from a compact Riemannian manifold \( M \) into the moduli space \( \mathcal{M}_1 \) of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z}) \backslash \mathbb{H} \). We prove that the flow is stable with respect to the energy functional and exhibits ergodic behavior in the sense that the evolving maps asymptotically distribute their image uniformly across the moduli space. As a concrete contribution, we show that the sequence of pushforward measures under the flow converges weak--$^{*}$ to the normalized hyperbolic measure on \( \mathcal{M}_1 \). Moreover, we introduce a relative entropy framework to measure the statistical deviation of the flow from equilibrium and prove that the relative entropy with respect to the hyperbolic measure decays to zero in the long-time limit. This provides a quantitative refinement of the ergodic result and establishes a connection between geometric flows, moduli space dynamics, and information-theoretic convergence.

Ergodic and Entropic Behavior of the Harmonic Map Heat Flow to the Moduli Space of Flat Tori

TL;DR

The paper analyzes the harmonic map heat flow from a compact manifold into the moduli space of unit-area flat tori, equipped with its natural hyperbolic geometry. It shows that the flow is energy-dissipative and converges to a harmonic map, while the pushforward measures of the evolving maps converge weak-* to the hyperbolic measure on ; the long-time distribution of the image is statistically uniform across the moduli space. A relative-entropy framework is developed, proving , which provides a quantitative refinement of ergodicity and links geometric flow dynamics with information-theoretic convergence. The work integrates harmonic map flow techniques with Teichmüller dynamics and entropy methods to connect geometric analysis, moduli-space dynamics, and probabilistic convergence in a rigorous, quantitative fashion.

Abstract

We investigate the harmonic map heat flow from a compact Riemannian manifold into the moduli space of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z}) \backslash \mathbb{H} \). We prove that the flow is stable with respect to the energy functional and exhibits ergodic behavior in the sense that the evolving maps asymptotically distribute their image uniformly across the moduli space. As a concrete contribution, we show that the sequence of pushforward measures under the flow converges weak-- to the normalized hyperbolic measure on . Moreover, we introduce a relative entropy framework to measure the statistical deviation of the flow from equilibrium and prove that the relative entropy with respect to the hyperbolic measure decays to zero in the long-time limit. This provides a quantitative refinement of the ergodic result and establishes a connection between geometric flows, moduli space dynamics, and information-theoretic convergence.
Paper Structure (9 sections, 7 theorems, 59 equations)

This paper contains 9 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. The harmonic map heat flow, backgrounds and the recent developments
  3. Moduli space of the flat tori and its geometry
  4. Preliminaries in Tiechmüller dynamics and hyperbolic geometry
  5. Hyperbolic geometry and hyperbolic Laplacian
  6. Teichmüller flows and its ergodic behavior
  7. General setting, Stability and convergence behavior
  8. Ergodic behavior
  9. Entropy in geometric analysis and dynamical systems

Key Result

Proposition 3.1

Let $\left(M,g\right)$ be a smooth compact Riemannian manifold without boundary, and let $\left(\mathcal{M}_{1},h\right)$ be the moduli space of flat tori with area one, equipped with the hyperbolic metric $h$. Let $\phi: M \times \left[0, \infty\right) \rightarrow \mathcal{M}_{1}$ be a smooth solut

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • Remark 4.4
  • Theorem 5.1
  • proof
  • Lemma 5.2
  • ...and 2 more