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Invariant distributions and the transport twistor space of closed surfaces

Jan Bohr, Thibault Lefeuvre, Gabriel P. Paternain

Abstract

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which play an important role in tensor tomography on surfaces - form a unital algebra, that is, multiplication of such distributions is well-defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Eventually, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio, asserting that invariant distributions of the geodesic flow of a positively-curved metric on the 2-sphere are determined by their zeroth and first Fourier modes.

Invariant distributions and the transport twistor space of closed surfaces

Abstract

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which play an important role in tensor tomography on surfaces - form a unital algebra, that is, multiplication of such distributions is well-defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Eventually, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio, asserting that invariant distributions of the geodesic flow of a positively-curved metric on the 2-sphere are determined by their zeroth and first Fourier modes.
Paper Structure (32 sections, 31 theorems, 158 equations, 1 figure, 1 table)

This paper contains 32 sections, 31 theorems, 158 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Algebras of holomorphic functions
  3. Regularity of invariant distributions on convex surfaces
  4. Holomorphic line bundles over twistor space
  5. Organisation of the article
  6. Geometry and analysis on the unit tangent bundle
  7. Unit tangent bundle
  8. Geometry of the unit tangent bundle
  9. Fourier decomposition
  10. Projective dynamics on the characteristic set
  11. Twist property of the vertical bundle
  12. Positive curvature
  13. Non conjugate points and nonpositive curvature
  14. Twistor space of surfaces
  15. Definition
  16. ...and 17 more sections

Key Result

Theorem 1.1

There is a well-defined trace map $f\mapsto f\vert_{SM}$ from $\mathcal{A}_\mathrm{pol}(Z)$ into $\mathcal{D}'(SM)$, such that is an isomorphism. In particular, the space of fibrewise holomorphic invariant distributions forms a unital algebra.

Figures (1)

  • Figure 1: The lifted dynamics on $\Sigma$. Left picture: dynamics near a point where $E_s^*$ and $E_u^*$ are transverse; the light blue region represents the cone $\mathcal{C}$. Right picture: dynamics over the set where $E_s^*=E_u^*=\mathbb{V}^*$; the cone $\mathcal{C}$ collapses here to a line.

Theorems & Definitions (66)

  • Theorem 1.1: Twistor correspondence for invariant distributions
  • Theorem 1.2: Computation of some algebras of holomorphic functions
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 56 more