Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Abstract

Let $(X,g)$ be a compact manifold with boundary $M^n$ and $\sigma$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincar\'e-Einstein metrics. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $\sigma^{-2}g$. We prove that if $\sigma$ is an approximate solution of a singular Yamabe problem, these families can be written as compositions of certain degenerate Laplacians. This result implies that the notions of extrinsic conformal Laplacians and extrinsic $Q$-curvature introduced in recent works by Gover and Waldron can naturally be rephrased in terms of residue families. The new spectral theoretical perspective enables us to relate the extrinsic conformal Laplacians and the critical extrinsic $Q$-curvature to the scattering operator of the asymptotically hyperbolic metric $\sigma^{-2}g$ extending the work of Graham and Zworski. The latter relation implies that the extrinsic conformal Laplacians are self-adjoint. We describe the asymptotic expansion of the volume of a singular Yamabe metric in terms of Laplace-Robin operators. We also derive new local holographic formulas for all extrinsic $Q$-curvatures in terms of renormalized volume coefficients, the scalar curvature of the background metric, and the asymptotic expansions of eigenfunctions of the Laplacian of the singular metric $\sigma^{-2}g$. Furthermore, we prove a new formula for the singular Yamabe obstruction $B_n$, and we use the latter formula to derive explicit expressions for the obstructions in low-order cases (confirming earlier results). Finally, we relate the obstruction $B_n$ to the supercritical $Q$-curvature $Q_{n+1}$.