Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomological equation for geodesic flows on flat surfaces

Giovanni Forni, Nelson Moll

TL;DR

This work establishes the existence and regularity of solutions to the cohomological equation for the flat geodesic flow on compact flat surfaces with cone points, under simultaneous Diophantine holonomy. It develops a harmonic-analysis framework on flat surfaces, analyzes the horizontal (holonomy) foliation, and derives a Cheeger-type bound for the foliated Laplacian, yielding hypoellipticity with codimension-one obstructions. The results demonstrate ergodicity of the holonomy foliation in the non-rational case and prove distributional solvability for zero-average data, with smooth solutions existing precisely when data lie in the kernel of the invariant-distribution space. Collectively, these findings extend Forni’s translation-surface methods to flat surfaces with non-rational holonomy and establish stability of the flat geodesic flow in the Katok sense.

Abstract

We prove the existence of solutions of the cohomological equation for the geodesic flow on the unit tangent bundle of a compact flat surface with finitely many cone points. We also prove the ergodicity of the holonomy foliation for surfaces with non-rational holonomy, and the cohomology-free property of the horizontal foliated Laplacian under a simultaneous Diophantine condition.

Cohomological equation for geodesic flows on flat surfaces

TL;DR

This work establishes the existence and regularity of solutions to the cohomological equation for the flat geodesic flow on compact flat surfaces with cone points, under simultaneous Diophantine holonomy. It develops a harmonic-analysis framework on flat surfaces, analyzes the horizontal (holonomy) foliation, and derives a Cheeger-type bound for the foliated Laplacian, yielding hypoellipticity with codimension-one obstructions. The results demonstrate ergodicity of the holonomy foliation in the non-rational case and prove distributional solvability for zero-average data, with smooth solutions existing precisely when data lie in the kernel of the invariant-distribution space. Collectively, these findings extend Forni’s translation-surface methods to flat surfaces with non-rational holonomy and establish stability of the flat geodesic flow in the Katok sense.

Abstract

We prove the existence of solutions of the cohomological equation for the geodesic flow on the unit tangent bundle of a compact flat surface with finitely many cone points. We also prove the ergodicity of the holonomy foliation for surfaces with non-rational holonomy, and the cohomology-free property of the horizontal foliated Laplacian under a simultaneous Diophantine condition.
Paper Structure (14 sections, 24 theorems, 136 equations)

This paper contains 14 sections, 24 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Geometry and Analysis on Flat Surfaces
  4. Local homogeneous structure
  5. A Fourier decomposition
  6. Local coordinates
  7. Sobolev spaces
  8. Creation and annihilation
  9. The horizontal (holonomy) foliation
  10. Ergodicity
  11. A Cheeger-type bound
  12. The cohomological equation
  13. Distributional and Smooth Solutions
  14. Invariant distributions

Key Result

Theorem 1.2

Suppose that $(R,S)$ has non-rational holonomy (a condition that holds in particular if at least one of the cone angles is irrational). Then the horizontal foliation $\mathcal{F}_H$ is ergodic, that is, any square integrable function on $M$ constant along almost every leaf of $\mathcal{F}_H$ is almo

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 36 more