Local rigidity of group actions of isometries on compact Riemannian manifolds

TL;DR

The paper establishes smooth and analytic local rigidity for finitely presented group actions by isometries on compact Riemannian manifolds under small perturbations. It builds a KAM-type iterative scheme controlled by $d_0$- and ${\square}$-Diophantine conditions to solve cohomological equations and reduce perturbations, with obstructions governed by the first cohomology $H^{1}(G,L^{2}(M,TM))$. In the analytic setting, Grauert tubes and Hardy spaces enable an analytic KAM scheme that yields convergence to a real-analytic conjugacy. Key corollaries include Fisher–Margulis-type results for groups with property (T) and rigidity for cyclic and abelian actions under Diophantine hypotheses. The work thus extends classical circle-diffeomorphism rigidity to higher dimensions and broad groups, linking spectral geometry, cohomology, and nonuniform small-divisor estimates to local rigidity phenomena.

Abstract

In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$. We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition. Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis's theorem on group actions satisfying Kazhdan's property (T).

Theorems & Definitions (67)

• Theorem 1.1
• Definition 1.2
• Remark 1.3
• Definition 1.4
• Remark 1.5
• Remark 1.6
• Definition 1.7
• Remark 1.8
• Definition 1.9
• Remark 1.10