Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cosymplectic Chern--Hamilton conjecture

Søren Dyhr, Ángel González-Prieto, Eva Miranda, Daniel Peralta-Salas

TL;DR

We address the Chern–Hamilton energy problem on compact cosymplectic 3-manifolds and classify all manifolds that admit a critical compatible metric. The main result shows that such a metric exists precisely when the Reeb flow is Killing (co-Kähler case) or when the manifold is a suspension cosymplectic mapping torus of $T^2$ by a hyperbolic automorphism, with the associated metric yielding a strictly positive, constant torsion and a minimal energy. The analysis connects the critical-point equation to $ abla_R igl( abla_R gigr)=0$ (or its cosymplectic reduction $ abla_R igl( abla_R gigr)=0$), and employs algebraic Anosov flows of Sol type to characterize the positive-torsion case, proving a global energy-minimization property. The work also identifies non-critical cosymplectic manifolds (including non-formal examples with $b_1\ge 2$) and establishes the equivalence between Sol-quotient and mapping-torus models for the critical metrics, highlighting deep connections between cosymplectic geometry, Anosov dynamics, and 3-manifold topology.

Abstract

In this paper, we study the Chern-Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-Kähler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number $b_1 \geq 2$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.

Cosymplectic Chern--Hamilton conjecture

TL;DR

We address the Chern–Hamilton energy problem on compact cosymplectic 3-manifolds and classify all manifolds that admit a critical compatible metric. The main result shows that such a metric exists precisely when the Reeb flow is Killing (co-Kähler case) or when the manifold is a suspension cosymplectic mapping torus of by a hyperbolic automorphism, with the associated metric yielding a strictly positive, constant torsion and a minimal energy. The analysis connects the critical-point equation to (or its cosymplectic reduction ), and employs algebraic Anosov flows of Sol type to characterize the positive-torsion case, proving a global energy-minimization property. The work also identifies non-critical cosymplectic manifolds (including non-formal examples with ) and establishes the equivalence between Sol-quotient and mapping-torus models for the critical metrics, highlighting deep connections between cosymplectic geometry, Anosov dynamics, and 3-manifold topology.

Abstract

In this paper, we study the Chern-Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-Kähler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.
Paper Structure (17 sections, 26 theorems, 82 equations, 1 table)

This paper contains 17 sections, 26 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples of cosymplectic manifolds
  4. Algebraic Anosov flows
  5. Compatible metrics and the Chern--Hamilton functional
  6. Equivalent models of hyperbolic mapping tori
  7. The critical point condition
  8. Properties and structure of critical metrics
  9. Proof of the main theorem
  10. Critical metrics with non-vanishing torsion
  11. Equivalence of models for critical metrics
  12. Cosymplectic manifolds without critical metrics
  13. Critical metrics globally minimize the energy
  14. Calculations for the critical point condition of the Chern--Hamilton energy functional
  15. Properties satisfied by compatible metrics
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(M, \alpha, \beta)$ be a compact, connected, $3$-dimensional cosymplectic manifold and let $R$ be its associated Reeb vector field. Then a compatible metric $g$ is critical for the Chern--Hamilton energy functional if and only if either Additionally, any critical metric minimizes the energy among all compatible metrics.

Theorems & Definitions (59)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Example 2.8
  • ...and 49 more