On singular Yamabe obstructions (Q2117495)

This paper is concerned with the obstruction to the existence of a (smooth) solution of a singular Yamabe problem for a hypersurface in a 4-dimensional Riemannian background geometry. This obstruction is a conformally invariant (weighted rank-0 tensor) density. Let \((X,g)\) be a compact \((n+1)\)-dimensional Riemannian manifold with \(n\)-dimensional boundary \(M\). The \emph{singular Yamabe problem} is to find a defining function \(\sigma\) of \(M\) such that \[ R(\sigma^{-2}g)=-n(n+1) \] where \(R(h)\) means the scalar curvature of the metric \(h\) on \(X\). Using the conformal transformation law of the scalar curvature and the trace \(J=\frac{R(g)}{2n}\) of the Schouten curvature of \(g\), the authors obtain the explicit differential equation \[ \mathcal S(g,\sigma) := |d\sigma|_g^2-\frac{2\sigma(\Delta_g(\sigma)+\sigma J)}{n+1} = 1\,. \] \smallskip Now the authors use geodesic normal coordinates (with \(r\) denoting the distance function of \(M\) for the background metric \(g\), i.e., \(g=dr^2+h_r\) with a 1-parameter family \(h_r\) of metrics on \(M\)) and introduce the defining function (\(\sigma_{(k)}\in C^\infty(M)\)) \[ \sigma_F = r+\sigma_{(2)}r^2+\dots+\sigma_{(n+1)}r^{n+1}\,, \] which satisfies \[ \mathcal S(g,\sigma) = 1+Rr^{n+1} \] where \(R\) is a smooth remainder term.\smallskip Following \textit{L. Andersson} et al. [Commun. Math. Phys. 149, No. 3, 587--612 (1992; Zbl 0764.53027)], the \emph{singular Yamabe obstruction} is defined by \[ \mathcal B_n := \left.\left( r^{-(n+1)}(\mathcal S(g,\sigma_F)-1) \right)\right|_{r=0}\,. \] It deserves to be mentioned that \(\mathcal B_n\) is conformally invariant of weight \(-(n+1)\).\medskip The cases \(\mathcal B_2\) and \(\mathcal B_3\) are discussed in detail by the authors.