## Scalar curvature rigidity of asymptotically hyperbolic spin manifolds.(English)Zbl 0686.53038

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.
Reviewer: Gh.Pitiş

### MSC:

 53C20 Global Riemannian geometry, including pinching
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### References:

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