Scalar curvature rigidity of asymptotically hyperbolic spin manifolds
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.
strongly asymptotically hyperbolic spin manifold
scientific article; zbMATH DE number 4124529