Boundary Curvature Scalars on Conformally Compact Manifolds
A. Rod Gover, Jarosław Kopiński, Andrew Waldron
TL;DR
This work develops a hierarchy of conformally invariant boundary curvature scalars for conformally compact manifolds, defined along the conformal boundary to quantify how CC metrics fail to solve the singular Yamabe problem. Using the Laplace–Robin operator and a carefully engineered sequence of delta_k operators, the authors extract canonical expansion coefficients of singular Yamabe metrics (and their Neumann data) as boundary invariants, with pole residues yielding obstructions such as generalized Willmore invariants. They construct a Dirichlet-to-Neumann map in criticial and restricted settings (superumbilic embeddings) and provide explicit formulae for the first five CC boundary scalars, along with their expansion coefficients and higher-order boundary operators. Their framework unifies expansion theory (Graham–Lee, Graham–Lee scales) with conformal boundary geometry, yielding both local invariants and a renormalized DTN map in four dimensions, with potential applications to asymptotically de Sitter spacetimes and renormalized volumes. The results advance the program of fully intrinsic, conformally invariant boundary data for singular Yamabe-type problems and illuminate links to Willmore-type variational invariants and Dirichlet-to-Neumann phenomena in geometric analysis and mathematical physics.
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.