×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math.36, 661-693 (1986) · Zbl 0598.53045
[2] Gromov, M., Lwason, H.B.: Spin and scalar curvature in the presence of a fundamental group I. Ann. Math.111, 209-230 (1980) · Zbl 0445.53025
[3] Gromov, M., Lawson, H.B.: The classification of simply-connected manifolds of positive scalar curvature. Ann. Math.111, 423-486 (1980) · Zbl 0463.53025
[4] Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math., Inst. Hautes. Etud. Sci.58, 295-408 (1983) · Zbl 0538.53047
[5] Hitchin, N.: Harmonic spinors. Adv. Math.14, 1-55 (1974) · Zbl 0284.58016
[6] Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris, Ser. A-B,25, 7-9 (1963) · Zbl 0136.18401
[7] Min-Oo, M.: Almost symmetric spaces. Preprint, McMaster University, 1987 · Zbl 0675.53044
[8] Min-Oo, M., Ruh, E.A.: Curvature deformations. In: Shiohama, K., Sakai, T., Sunada, T. (eds.) Curvature and topology of Riemannian manifolds. Proceedings, Katata 1985. (Lect. Notes Math., vol. 1201, pp. 122-133) Berlin Heidelberg New York: Springer 1986 · Zbl 0634.53030
[9] Müller, W.: Manifolds with cusps of rank one. Lect. Notes Math. 1244. Berlin Heidelberg New York: Springer 1987 · Zbl 0632.58001
[10] Parker, T.H., Taubes, C.: On Witten’s proof of the positive energy theorem, Commun. Math. Phys.84, 223-238 (1982) · Zbl 0528.58040
[11] Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity Commun. Math. Phys.65, 45-76 (1979) · Zbl 0405.53045
[12] Schoen, R., Yau, S.T.: Proof of the positive action conjecture in quantum relativity. Phys. Rev. Lett.42, 547-548 (1979)
[13] Schoen, R., Yau, S.T.: Proof of the positive mass theorem II. Commun. Math. Phys.79, 231-260 (1981) · Zbl 0494.53028
[14] Schroeder, V., Ziller, W.: Local rigidity symmetric spaces. Preprint Inst. Hautes. Etud. Sci. (1987) · Zbl 0724.53033
[15] Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys.80, 381-402 (1981) · Zbl 1051.83532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.