A new proof of Lee's theorem on the spectrum of conformally compact Einstein manifolds
Let \(M\) be a compact \((m+1)\)-dimensional manifold with boundary \(\Sigma\). One says a Riemannian metric on \(\text{int}(M)\) is conformally compact if for any defining function \(r\) of the boundary, \(\tilde g:=r^2g\) extends to a \(C^3\) metric on all of \(M\). The conformal class of \(\tilde g _\Sigma\) is invariantly defined and is called the conformal infinity of \(g\). Mazzeo showed the spectrum of such a manifold is the ray \([m^2/4,\infty)\) with no embedded eigenvalues -- there can be finitely many eigenvalues in the interval \((0,n^2/4)\). \textit{J. Lee} [Commun. Anal. Geom. 3, 253-271 (1995; Zbl 0934.58029)] showed that if the conformal infinity has non-negative Yamabi invariant, then \(\lambda_0(g)=n^2/4\). In the present paper, the author uses ideas of Witten-Yau and Cai-Galloway to give a simple proof of this result.
scientific article; zbMATH DE number 1893859