Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1907.01809

Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools

Yannick Guedes Bonthonneau·Thibault Lefeuvre
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is the first in a series of two articles whose aim is to extend a recent result of Guillarmou-Lefeuvre on the local rigidity of the marked length spectrum from the case of compact negatively-curved Riemannian manifolds to the case of manifolds with hyperbolic cusps. In this first paper, we deal with the linear (or infinitesimal) version of the problem and prove that such manifolds are spectrally rigid for compactly supported deformations. More precisely, we prove that the X-ray transform on symmetric solenoidal 2-tensors is injective. In order to do so, we expand the microlocal calculus developed by Bonthonneau and Bonthonneau-Weich to be able to invert pseudodifferential operators on Sobolev and Hölder-Zygmund spaces modulo compact remainders. This theory has an interest on its own and is extensively used in the second paper [arXiv:1910.02154] in order to deal with the nonlinear problem.

Explore related subjects

math.DGmath.APmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yannick Guedes Bonthonneau, Thibault Lefeuvre. 2020-11-26. Local rigidity of manifolds with hyperbolic cusps I. Linear theory and microlocal tools. https://arxiv.org/abs/1907.01809

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the topology of manifolds with nonnegative Ricci curvature and linear volume growth

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and linear volume growth. First, we show that the fundamental group of such a manifold contains a subgroup $\mathbb{Z}^k$ of finite index, where $0\le k\le n-1$. Second, we prove that if the Ricci curvature is positive everywhere, then the fundamental group is finite. The proofs are based on an analysis of the equivariant asymptotic geometry of successive covering spaces and a plane/halfplane rigidity result for RCD spaces.

math.DG↗
arXiv2026

K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers

Recently, Sun-Zhang have developed an algebraic theory for Kähler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(π: X \to Y, ξ)$. In particular, they conjecture that existence of a Kähler-Ricci shrinker metric is equivalent to a notion of K-stability. We prove one direction of this conjecture, namely that existence of a Kähler-Ricci shrinker metric $g$ implies K-polystability of $(π: X \to Y, ξ)$, in the case that the Ricci curvature of $g$ decays at infinity. As an application, we give a non-existence result: if $M$ is the blowup of a six-dimensional quadric along a two-dimensional subquadric, then the total space $X$ of the cube root of $K_M$ is a polarized Fano fibration not admitting a Kähler-Ricci shrinker.

math.DG↗
arXiv2026

Observações sobre funções potenciais de variedades quase-Einstein não compactas

Neste artigo, estudamos o conjunto de funções potenciais em variedades quase Einstein não compactas. Mostramos que o espaço de todas as funções potenciais positivas em uma variedade tridimensional não compacta quase-Einstein tem dimensão no máximo dois, e que a igualdade vale se e somente se a variedade for isométrica a um produto $B\times\mathbb{R}$, onde $B$ é uma superfície $λ$-Einstein ou um dos exemplos obtidos por L. Berard Bergery e descritos no livro de Besse. Além disso, provamos que qualquer variedade quase-Einstein assintoticamente plana $n$-dimensional com $λ=0$ é necessariamente Ricci-plana.

math.DG↗