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Quantitative marked length spectrum rigidity

Karen Butt

TL;DR

This work develops a quantitative version of marked length spectrum rigidity in negative curvature by combining Liouville-current analysis, cross-ratio controls, and the Besson–Courtois–Gallot framework. It establishes explicit ε-dependent volume and derivative bounds for the BCG map when MLS of two manifolds are ε-close, leading to near isometry in higher dimensions. A parallel argument for compact negatively curved surfaces, using Otal and Pugh techniques, yields a distance-preserving correspondence in the limit. Collectively, the results provide a rigorous, quantitative stability theory for marked length spectrum rigidity and its geometric consequences.

Abstract

We consider a closed Riemannian manifold $M$ of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space $N$. Using the methods of Hamenstädt, we show the volumes of $M$ and $N$ are approximately equal. We then show the Besson-Courtois-Gallot map $F: M \to N$ is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson-Courtois-Gallot, which show $M$ and $N$ are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh.

Quantitative marked length spectrum rigidity

TL;DR

This work develops a quantitative version of marked length spectrum rigidity in negative curvature by combining Liouville-current analysis, cross-ratio controls, and the Besson–Courtois–Gallot framework. It establishes explicit ε-dependent volume and derivative bounds for the BCG map when MLS of two manifolds are ε-close, leading to near isometry in higher dimensions. A parallel argument for compact negatively curved surfaces, using Otal and Pugh techniques, yields a distance-preserving correspondence in the limit. Collectively, the results provide a rigorous, quantitative stability theory for marked length spectrum rigidity and its geometric consequences.

Abstract

We consider a closed Riemannian manifold of negative curvature and dimension at least 3 with marked length spectrum sufficiently close (multiplicatively) to that of a locally symmetric space . Using the methods of Hamenstädt, we show the volumes of and are approximately equal. We then show the Besson-Courtois-Gallot map is a diffeomorphism with derivative bounds close to 1 and depending on the ratio of the two marked length spectrum functions. Thus, we refine the results of Hamenstädt and Besson-Courtois-Gallot, which show and are isometric if their marked length spectra are equal. We also prove a similar result for compact negatively curved surfaces using the methods of Otal together with a version of the Gromov compactness theorem due to Pugh.
Paper Structure (21 sections, 65 theorems, 181 equations)

This paper contains 21 sections, 65 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Volume estimate
  3. The cross-ratio
  4. The Liouville current
  5. $E_v$-cubes
  6. Quasi-symplecticity
  7. A controlled orbit equivalence
  8. A controlled quasi-isometry
  9. Combining the constructions of Bourdon and Gromov
  10. Proof of the volume estimate
  11. Estimates for the BCG map
  12. Construction of the BCG map
  13. Lipschitz bound for $F$
  14. Lipschitz constant for ${\rm Jac} F(p)$
  15. Lower bound for $|{\rm Jac} F(p)|$
  16. ...and 6 more sections

Key Result

Theorem A

Let $(M, g_0)$ and $\mathcal{U}_{g_0}^{1, \alpha} (a, b, R)$ as above. Fix $L > 1$. Then there exists $\varepsilon = \varepsilon(L, a, b, R, \alpha) > 0$ small enough so that for any pair $(M, g), (M, h) \in \mathcal{U}_{g_0}^{1, \alpha} (a, b, R)$ satisfying there exists an $L$-Lipschitz map $f: (M, g) \to (M, h)$.

Theorems & Definitions (139)

  • Theorem A
  • Theorem B
  • Remark 1.3
  • Theorem C
  • Remark 1.4
  • Remark 1.5
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Remark 2.5
  • ...and 129 more