Scalar flat compactifications of Poincaré-Einstein manifolds and applications

Raulot, Simon

doi:10.1090/ecgd/371

Scalar flat compactifications of Poincaré-Einstein manifolds and applications
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Conform. Geom. Dyn. 26 (2022), 46-66 Request permission

Abstract:

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincaré-Einstein manifolds. As a first consequence, we obtain a sharp lower bound for the first eigenvalue of the conformal half-Laplacian of the boundary of such manifolds. Secondly, a new upper bound for the renormalized volume is given in the four dimensional setting. Finally, some estimates on the first eigenvalues of Dirac operators are also deduced.

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