Boundary Curvature Scalars on Conformally Compact Manifolds

TL;DR

This paper introduces conformally invariant scalar curvature quantities on conformally compact manifolds that measure deviations from constant negative scalar curvature, providing tools for analyzing singular Yamabe problems and related geometric invariants.

Contribution

It defines new boundary curvature scalars that are conformally invariant and relate to the singular Yamabe problem, offering explicit formulas and connections to geometric invariants.

Findings

Defined conformally invariant boundary curvature scalars.

Derived residues as obstructions to smooth solutions.

Provided explicit formulas for key scalars in four dimensions.

Abstract

We introduce a sequence of conformally invariant scalar curvature quantities, defined along the conformal infinity of a conformally compact (CC) manifold, that measure the failure of a CC metric to have constant negative scalar curvature in the interior, i.e. its failure to solve the singular Yamabe problem. Indeed, these "CC boundary curvature scalars" compute canonical expansion coefficients for singular Yamabe metrics. Residues of their poles yield obstructions to smooth solutions to the singular Yamabe problem and thus, in particular, give an alternate derivation of generalized Willmore invariants. Moreover, in a given dimension, the critical CC boundary scalar characterizes the image of a Dirichlet-to-Neumann map for the singular Yamabe problem. We give explicit formulae for the first five CC boundary curvature scalars required for a global study of four dimensional singular Yamabe metrics, as well as asymptotically de Sitter spacetimes.