Scalar Flat Compactifications Of Poincar{\'e}-einstein Manifolds And Applications

TL;DR

This paper establishes new inequalities relating boundary curvature and scalar curvature in Poincaré-Einstein manifolds, leading to bounds on eigenvalues and volume, with implications for geometric analysis.

Contribution

It introduces novel integral inequalities and bounds for eigenvalues and volume in scalar flat conformal compactifications of Poincaré-Einstein manifolds.

Findings

Sharp lower bound for the first eigenvalue of the conformal half-Laplacian

New upper bound for the renormalized volume in four dimensions

Estimates on the first eigenvalues of Dirac operators

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{\'e}-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.