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Katok's entropy conjecture near real and complex hyperbolic metrics

Tristan Humbert

TL;DR

The paper addresses Katok's entropy rigidity in a nonconformal setting by proving a local rigidity result near locally symmetric metrics. It develops a transverse (solenoidal) analysis of metric deformations using the generalized X-ray transform $\Pi$ and an explicit differential operator $Q$, enabling a Hessian-based stability bound for the entropy gap functional $\Phi(g)=Ent_{\mathrm{top}}(g)-Ent_{\mathrm{Liou}}(g)$. The main theorem shows that near a real or complex hyperbolic metric $g_0$, entropy equality together with equal volume implies $g$ is isometric to $g_0$ (i.e., locally symmetric), with extensions to other locally symmetric types contingent on solenoidal injectivity. The work yields local rigidity results for hyperbolic rank and for metrics with $C^2$ Anosov foliations, and relies on Pestov-type identities, Weitzenböck formulas, and a microlocal ellipticity/injectivity analysis of $\Pi_{\ker(D_{g_0}^*)}Q$ in the real and complex hyperbolic settings. The approach blends geometric analysis on the unit tangent bundle with microlocal and representation-theoretic tools (raising/lowering operators) to connect dynamical entropy with differential-geometric rigidity, offering a path toward entropy rigidity near all locally symmetric spaces.

Abstract

We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with $C^2$ Anosov foliations near complex hyperbolic metrics.

Katok's entropy conjecture near real and complex hyperbolic metrics

TL;DR

The paper addresses Katok's entropy rigidity in a nonconformal setting by proving a local rigidity result near locally symmetric metrics. It develops a transverse (solenoidal) analysis of metric deformations using the generalized X-ray transform and an explicit differential operator , enabling a Hessian-based stability bound for the entropy gap functional . The main theorem shows that near a real or complex hyperbolic metric , entropy equality together with equal volume implies is isometric to (i.e., locally symmetric), with extensions to other locally symmetric types contingent on solenoidal injectivity. The work yields local rigidity results for hyperbolic rank and for metrics with Anosov foliations, and relies on Pestov-type identities, Weitzenböck formulas, and a microlocal ellipticity/injectivity analysis of in the real and complex hyperbolic settings. The approach blends geometric analysis on the unit tangent bundle with microlocal and representation-theoretic tools (raising/lowering operators) to connect dynamical entropy with differential-geometric rigidity, offering a path toward entropy rigidity near all locally symmetric spaces.

Abstract

We show that, given a real or complex hyperbolic metric on a closed manifold of dimension , there exists a neighborhood of in the space of negatively curved metrics such that for any , the topological entropy and Liouville entropy of coincide if and only if and are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with Anosov foliations near complex hyperbolic metrics.
Paper Structure (27 sections, 19 theorems, 222 equations, 1 figure)

This paper contains 27 sections, 19 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Existing results
  4. Entropy rigidity near locally symmetric metrics
  5. Local hyperbolic rank rigidity
  6. Local rigidity for metric with $C^2$ stable and unstable foliations.
  7. Strategy of the proof.
  8. Preliminaries
  9. Symmetric tensors and analysis on $SM$.
  10. Tensorial analysis
  11. Analysis on $SM$
  12. Symmetrized covariant derivative and geodesic vector field.
  13. Pestov's identity
  14. Decomposition of the space of symmetric tensors.
  15. The operator $\Pi$
  16. ...and 12 more sections

Key Result

Theorem 1

Let $(M^n,g_0)$ be a closed manifold of dimension $n\geq 3$ and suppose that $g_0$ is either a real hyperbolic or a complex hyperbolic metric. Then there exists $N(n)\in \mathbb N$ and $\epsilon>0$ such that for any negatively curved metric $g$ on $M$ with $\|g-g_0\|_{C^N}<\epsilon$, if one has $\ma

Figures (1)

  • Figure 1: An illustration of the slice Lemma \ref{['slice lemma']} in a neighborhood $\mathcal{U}$ of $g_0$.

Theorems & Definitions (37)

  • Conjecture 1: Katok
  • Theorem 1: Entropy rigidity near real and complex hyperbolic spaces
  • Theorem 2
  • Theorem 3: Solenoidal injectivity
  • Conjecture 2: Connell-Nguyen-Spatzier
  • Theorem 4: Local rigidity of metric with $C^2$ foliation near real and complex hyperbolic metrics
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3: Slice lemma
  • ...and 27 more