Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917128666349568 |
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| author | Butt, Karen Erchenko, Alena Humbert, Tristan Lefeuvre, Thibault Wilkinson, Amie |
| author_facet | Butt, Karen Erchenko, Alena Humbert, Tristan Lefeuvre, Thibault Wilkinson, Amie |
| contents | 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 $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3 Butt, Karen Erchenko, Alena Humbert, Tristan Lefeuvre, Thibault Wilkinson, Amie Differential Geometry Dynamical Systems Spectral Theory 37C27, 37D40, 53C22, 53C24, 53C35 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 $g_0$ and both metrics have the same MPD, then they are homothetic. As a by-product of our proof, we show the Lichnerowicz Laplacian of $g_0$ 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. |
| title | Marked Poincaré rigidity near hyperbolic metrics and injectivity of the Lichnerowicz Laplacian in dimension 3 |
| topic | Differential Geometry Dynamical Systems Spectral Theory 37C27, 37D40, 53C22, 53C24, 53C35 |
| url | https://arxiv.org/abs/2510.11399 |