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Length spectrum rigidity and flexibility of spheres of revolution with one equator

Alberto Abbondandolo, Marco Mazzucchelli

TL;DR

This work studies length-spectrum rigidity for $S^1$-invariant metrics on $S^2$ with a single equator, introducing the marked length spectrum consisting of the equator length and the set of lengths of closed geodesics of each coprime type $(p,q)$. The authors develop a dynamical framework using the Birkhoff annulus and a generating function $F$ to relate the marked length spectrum to the first return data, proving that isospectral metrics have $S^1$-equivariant conjugacies away from the equator and, under positive curvature or nonnegative curvature near the equator, can extend (smoothly or continuously) to the whole unit tangent bundle. They establish a rigidity/flexibility dichotomy: under a $\ extbf{Z}_2$-symmetry assumption, the marked length spectrum determines the metric, while the isospectral set in the $\ extbf{Z}_2$-symmetric class is an infinite-dimensional convex set, generalizing Zoll-manifold descriptions. The appendices provide elementary proofs linking tangent lines to a graph with the generating function framework and classify a class of $S^1$-invariant contact forms, reinforcing the geometric structure behind the dynamical results.

Abstract

We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds.

Length spectrum rigidity and flexibility of spheres of revolution with one equator

TL;DR

This work studies length-spectrum rigidity for -invariant metrics on with a single equator, introducing the marked length spectrum consisting of the equator length and the set of lengths of closed geodesics of each coprime type . The authors develop a dynamical framework using the Birkhoff annulus and a generating function to relate the marked length spectrum to the first return data, proving that isospectral metrics have -equivariant conjugacies away from the equator and, under positive curvature or nonnegative curvature near the equator, can extend (smoothly or continuously) to the whole unit tangent bundle. They establish a rigidity/flexibility dichotomy: under a -symmetry assumption, the marked length spectrum determines the metric, while the isospectral set in the -symmetric class is an infinite-dimensional convex set, generalizing Zoll-manifold descriptions. The appendices provide elementary proofs linking tangent lines to a graph with the generating function framework and classify a class of -invariant contact forms, reinforcing the geometric structure behind the dynamical results.

Abstract

We define a notion of marked length spectrum for -symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further -symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique -symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of -symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of -invariant contact forms on three-manifolds.
Paper Structure (19 sections, 29 theorems, 281 equations)

This paper contains 19 sections, 29 theorems, 281 equations.

Table of Contents

  1. Proofs' ideas and organization of the paper.
  2. Acknowledgments
  3. Geodesic flows on $S^1$-symmetric spheres
  4. The Birkhoff annulus
  5. The generating function
  6. The marked length spectrum
  7. Construction of the conjugacy away from the equator
  8. Regular extension of the conjugacy at the equator
  9. Continuous extension of the conjugacy at the equator
  10. Proof of Theorem \ref{['thmB']} and order of vanishing of the curvature
  11. Isospectral classes
  12. Examples without smooth $S^1$-equivariant conjugacy
  13. Unstable equators
  14. Determining a function from the tangent lines to its graph
  15. $S^1$-invariant contact forms on three-manifolds
  16. ...and 4 more sections

Key Result

Theorem A

Let $g_1, g_2 \in \mathcal{G}$ be smooth $($resp. analytic$)$ isospectral metrics. Then there exists a smooth $($resp. analytic$)$$S^1$-equivariant contactomorphism which conjugates the geodesic flows of $g_1$ and $g_2$, i.e. Furthermore$:$

Theorems & Definitions (61)

  • Theorem A
  • Theorem B
  • Corollary C
  • Corollary D
  • Lemma 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • Remark 3.2
  • Remark 4.1
  • ...and 51 more