Scalar Flat Compactifications Of Poincar{é}-einstein Manifolds And Applications

Abstract

We derive an integral inequality between the mean curvature and the scalar curvature of the boundary of any scalar flat conformal compactifications of Poincar{é}-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.

References in corpus (3)

• The Hijazi inequality on manifolds with boundary
• On a spin conformal invariant on manifolds with boundary
• A remark on the rigidity of conformally compact Poincar{é}-Einstein manifolds