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Gap phenomena and curvature estimates for conformally compact Einstein manifolds. (English) Zbl 1364.53048

The authors present results on gap phenomena and curvature estimates for conformally compact Einstein manifolds (with implications in the theory of quantum gravity in theoretical physics). First, they present a gap theorem, using a blow-up method, for a class of conformally compact Einstein manifolds to derive curvature estimates for these manifolds with large renormalized volume. Other results are on conformally compact Einstein manifolds whose conformal infinity has a large positive Yamabe invariant (the renormalized volume is the geometric property of the bulk space \((X^4,g^+)\), while the Yamabe constant \(Y(\delta X, |\hat{g}|)\) is an invariant of the conformal infinity only).
The main focus of the paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. The authors also give a complete proof of the rigidity theorem for a conformally compact Einstein manifold (with conformal infinities having large Yamabe invariant) in arbitrary dimension without spin assumption with a new curvature pinching estimate. As a consequence, they also derive curvature estimates for these manifolds whose conformal infinity has a large Yamabe constant.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J05 Elliptic equations on manifolds, general theory
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