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STUDIES OF SOME CURVATURE OPERATORS IN A NEIGHBORHOOD OF AN ASYMPTOTICALLY

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STUDIES OF SOME CURVATURE OPERATORS IN A NEIGHBORHOOD OF AN ASYMPTOTICALLY HYPERBOLIC EINSTEIN MANIFOLD ERWANN DELAY Abstract. On an Asymptotically Hyperbolic Einstein manifold (M, g0) for which the Yamabe invariant of the conformal structure on the boundary at infinity is non-negative, we show that the op- erators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metric g0. We deduce in the C∞ case that the image of the Riemann-Christoffel curvature op- erator is a submanifold in a neighborhood of g0. Keywords : Einstein Manifold, Asymptotically hyperbolic; curvatures of Riemann-Christoffel, Ricci, Einstein ; nonlinear PDE, elliptic degen- erate, asymptotic behaviour. 2000 MSC : 58C15, 53C21, 35J70, 35B45, 35B40 1. Introduction Conformally compact manifolds form a very important class of non- compact manifolds either in Riemannian geometry or in General Rela- tivity. We are interested here in such manifolds which are asymptoti- cally hyperbolic (their sectional curvature approches?1 at infinity) and Einstein. We show that the Ricci curvature and the Einstein curvature can be arbitrarily prescribed on the manifold in the neighborhood of an Asymptotically Hyperbolic Einstein Metric (AHEM). We deduce that the Image of the Riemann-Christoffel operator is a submanifold near an AHEM. This result extends a previous work [D1] on the real hyperbolic space.

  • anti-symmetric tensors

  • ricci curvature

  • laplacian

  • tensors

  • ahem metric

  • hyperbolic einstein

  • einstein curvature

  • manifold

  • g0-conformal tensors


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STUDIES OF SOME CURVATURE OPERATORS IN A NEIGHBORHOOD OF AN ASYMPTOTICALLY HYPERBOLIC EINSTEIN MANIFOLD
ERWANN DELAY
Abstract.On an Asymptotically Hyperbolic Einstein manifold (M, g0) for which the Yamabe invariant of the conformal structure on the boundary at infinity is non-negative, we show that the op-erators of Ricci curvature, and of Einstein curvature, are locally invertible in a neighborhood of the metricg0. We deduce in the Ccase that the image of the Riemann-Christoffel curvature op-erator is a submanifold in a neighborhood ofg0.
Keywords: Einstein Manifold, Asymptotically hyperbolic; curvatures of Riemann-Christoffel, Ricci, Einstein ; nonlinear PDE, elliptic degen-erate, asymptotic behaviour.
2000 MSC: 58C15, 53C21, 35J70, 35B45, 35B40
1.Introduction Conformally compact manifolds form a very important class of non-compact manifolds either in Riemannian geometry or in General Rela-tivity. We are interested here in such manifolds which are asymptoti-cally hyperbolic (their sectional curvature approches1 at infinity) and Einstein. We show that the Ricci curvature and the Einstein curvature can be arbitrarily prescribed on the manifold in the neighborhood of an Asymptotically Hyperbolic Einstein Metric (AHEM). We deduce that the Image of the Riemann-Christoffel operator is a submanifold near an AHEM. This result extends a previous work [D1] on the real hyperbolic space. The method to invert the Ricci or Einstein opera-tor is an implicit function theorem in the neighborhood of the AHEM metriconsomeweightedHo¨lderspaces.Theproblemisthenreduced to invert an operator of kind4+Kon symmetric covariant two tensor where4is the Laplacian associated to the AHEM andKa term of order zero; for instance we have to invert the Lichnerowicz Laplacian in
DateResearch partially supported by the EU TMR project: February 20th 2001. Stochastic Analysis and its Applications, ERB-FMRX-CT96-0075. 1
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