Scalar curvature rigidity of asymptotically hyperbolic spin manifolds
type: Item
instance of: scholarly article
title: Scalar curvature rigidity of asymptotically hyperbolic spin manifolds
zbMATH Open document ID: 0686.53038
DOI: 10.1007/BF01452046
author: Maung Min-Oo
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164616
review text: A smooth Riemannian manifold \((M^ n,g)\) is strongly asymptotically hyperbolic if there exists a compact subset \(B\subset M^ n\) and a diffeomorphism \(\phi: M^ n\setminus B\to H^ n(r_ 0,\infty)\) for some \(r_ 0>0\), such that the gauge transformation \(A: T(M^ n\setminus B)\to T(M^ n\setminus B),\) defined by the conditions \[ g(Au,Av)=\phi^*g(u,v)=g(d\phi (u),d\phi (v)),\quad g(Au,v)=g(u,Av) \] satisfies the following properties: a) there exists a uniform Lipschitz constant \(C\geq 1\) such that \(C^{-1}\leq \min | Av| \leq \max | Av| \leq C\) for all \(v\in T(M^ n\setminus B)\) with \(| v| =1\); \[ b)\quad \exp (\phi \circ r).(A-id)\in L^{1,2}(T^*(M^ n\setminus B)\otimes T(M^ n\setminus B),g). \] The main result of this paper is the following: A strongly asymptotically hyperbolic spin manifold of dimension \(n\geq 3\), whose scalar curvature satisfies \(R\geq - n(n-1)\) everywhere, is isometric to the hyperbolic space \(H^ n\). For the proof of this theorem the author uses a Weitzenböck formula for the Dirac operator associated to a Cartan connection of hyperbolic type.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4124529
zbMATH Keywords: strongly asymptotically hyperbolic spin manifold
zbMATH Keywords: scalar curvature
zbMATH Keywords: hyperbolic space
zbMATH Keywords: Weitzenböck formula
zbMATH Keywords: Dirac operator
reviewed by: Gheorghe Pitiş
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124529