Renormalized volume and non-degeneracy of conformally compact Einstein four-manifolds (Q6564589)

Let \((X, g_{+})\) be a four-dimensional Poincaré-Einstein manifold with conformal infinity \((M, [\hat{g}])\), where \(M = \partial X\). Let \(L\) denote the (gauge-fixed) linearized Einstein operator acting on trace-free symmetric tensors:\N\[\N(Lh)_{ij} =-\Delta h_{ij} -2W_{ikjl}h_{kl}-2h_{ij} .\N\]\N\textit{J. M. Lee} [Fredholm operators and Einstein metrics on conformally compact manifolds. Providence, RI: American Mathematical Society (AMS) (2006; Zbl 1112.53002)] showed that if the \(L^{2}\)-kernel of \(L\) (with respect to \(g_+\)) is trivial, then for any sufficiently small perturbation of the conformal infinity \((M, [\hat{g}])\) there is a Poincaré-Einstein metric close to \(g_{+}\) whose conformal infinity is the given perturbation.\N\NThe main result of the paper is a conformally invariant condition in dimension four which implies nondegeneracy of the operator \(L\). This condition can be stated in two forms:\N\begin{itemize}\N\item[1.] As a smallness condition on the \(L^{2}\)-norm of the Weyl tensor of \(g_{+}\): if the Yamabe invariant of \((M, [\hat{g}])\) is positive and the Weyl tensor of \(g_{+}\) satisfies\N\[\N\int_{X} |W_{g_{+}}|^{2} dV_{g_{+}} \leq \frac{32}{9}\pi^{2}\chi(X) .\N\]\Nthen \(L\) is nondegenerate;\N\item[2.] In terms of the renormalized volume: if the Yamabe invariant of \((M, [\hat{g}])\) is positive and the renormalized volume \(V\) of \((X, g_{+})\) satisfies\N\[\NV\geq \frac{32}{27}\pi^{2}\chi(X) \ ,\N\]\Nthen \(L\) is nondegenerate.\N\end{itemize}