Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3

2025/10/13 by Butt, Karen, Erchenko, Alena, Humbert, Tristan +2 · 1 citation
#37C27 #37D40 #53C22 #53C24 #53C35 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2510.11399

Abstract

Let M be a compact manifold without boundary equipped with a Riemannian metric g of negative curvature. In this paper, we introduce the marked Poincaré determinant (MPD), a homothety invariant of g depending on differentiable periodic data of its geodesic flow. The MPD associates to each free homotopy class of closed curves in M a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. We prove a local MPD rigidity result in dimension 3: if g is sufficiently close to a hyperbolic metric g₀ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of g₀ is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

Citations

• Katok's entropy conjecture near real and complex hyperbolic metrics
• Marked length spectrum rigidity for Anosov surfaces
• Smooth rigidity for 3-dimensional volume preserving Anosov flows and weighted marked length spectrum rigidity
• Stability of Einstein metrics on symmetric spaces of compact type
• Radial source estimates in Hölder-Zygmund spaces for hyperbolic dynamics
• Abelian Livshits theorems and geometric applications
• Geometry in the large of the kernel of Lichnerowicz Laplacians and its applications
• The marked length spectrum of Anosov manifolds
• Characterizing Symmetric Spaces by their Lyapunov Spectra
• Rigidity of equality of Lyapunov exponents for geodesic flows
• On Contact Anosov Flows
• Cocycles, symplectic structures and intersection

Cited by

• Monotonicity of topological entropy along the Ricci flow near a hyperbolic metric

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• Rigidity results with curvature conditions from Lichnerowicz Laplacian and applications
• Determinants of Laplacians and Isopolar Metrics on Surfaces of Infinite Area
• Geometric and analytic structures on metric spaces homeomorphic to a manifold
• Rigidity of Totally Geodesic Hypersurfaces in Negative Curvature
• Quantitative marked length spectrum rigidity