Optimal coordinates for Ricci-flat conifolds

TL;DR

This paper analyzes the behavior of Ricci-flat conifolds by computing indicial roots of the Lichnerowicz Laplacian, providing convergence rates for metrics near conical ends, and clarifying properties of Ricci-flat ALE manifolds.

Contribution

It offers explicit calculations of indicial roots and convergence orders for Ricci-flat conifolds, filling a gap in previous understanding of ALE manifolds.

Findings

Computed indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones

Established lower bounds for metric convergence rates at conical ends

Proved Ricci-flat ALE manifolds are of order n

Abstract

We compute the indicial roots of the Lichnerowicz Laplacian on Ricci-flat cones and give a detailed description of the corresponding radially homogeneous tensor fields in its kernel. For a Ricci-flat conifold $(M,g)$ which may have asymptotically conical as well as conically singular ends, we compute at each end a lower bound for the order with which the metric converges to the tangent cone. As a special subcase of our result, we show that any Ricci-flat ALE manifold $(M^n,g)$ is of order $n$ and thereby close a small gap in a paper by Cheeger and Tian.