Optimal coordinates for Ricci-flat conifolds (Q6583652)

Let \(\widehat{M}^{n-1}\) be a closed manifold and let \(\overline{P}\) be a self-adjoint Laplace type operator on \(\overline{M}=(0,\infty)\times \widehat{M}\), which is of the form \N\[\N\overline{P}=-\partial_{rr}^2-\frac{n-2}{n}\partial_r+\frac{1}{r^2}\widehat{P},\N\]\Nfor some Laplace type operator \(\widehat{P}\) on \(\widehat{M}\). If \(\overline{P}\) is of such a form and \(\nu\in \mathbb{R}\) is an eigenvalue of \(\widehat{P}\), the values \N\[\N\xi_{\pm}(\nu):=-\frac{n-2}{2}\pm\sqrt{\frac{(n-2)^2}{4}+\nu} \N\]\Nare called indicial roots of \(\overline{P}\).\N\NThis paper focuses on computing the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and describing radially homogeneous tensor fields in its kernel. For Ricci-flat conifolds, a lower bound is calculated for the order with which the metric converges to the tangent cone at each end.\N\NThe main result of the paper is that it provides a detailed computation of the indicial roots and a formula for the optimal decay of the metric at each end of a Ricci-flat conifold. The results extend previous works and by showing that any Ricci-flat ALE manifold \((M^n,g)\) is of order \(n\), close a gap in a paper by \textit{J. Cheeger} and \textit{G. Tian} [Invent. Math. 118, No. 3, 493--571 (1994; Zbl 0814.53034)].\N\NThe main techniques used in the study can be summarized as follows. Commutation formulas involving the Lichnerowicz Laplacian and other operators are used to simplify the computation of indicial roots. Elliptic regularity is employed to handle the decay rates and regularity of solutions to elliptic partial differential equations on conifolds. The paper constructs optimal coordinates by imposing the Bianchi gauge, which helps in analyzing the asymptotic behavior of Ricci-flat metrics.\N\NThe results have potential applications in desingularizing of Einstein conifolds by smooth Einstein metrics and computing convergence rates of other geometric structures at infinity. It might be possible to analyze the stability of Ricci-flat conifolds under various geometric flows, such as the Ricci flow or mean curvature flow using these results. Understanding the formation and resolution of singularities in Einstein metrics and their relation to the indicial roots of the Lichnerowicz Laplacian can be another route for future research where this study may be exploited.