Geodesic Equations on asymptotically locally Euclidean K\"ahler manifolds

TL;DR

This paper solves the geodesic equation in the space of K"ahler metrics on ALE manifolds, establishes regularity of solutions, and explores implications for scalar-flat metrics and Ricci curvature conditions.

Contribution

It provides the first global regularity results for geodesics in the space of K"ahler metrics on ALE manifolds and links these solutions to scalar-flat metric uniqueness.

Findings

Established global ,1 regularity of geodesic solutions.

Connected geodesic solutions to the uniqueness of scalar-flat ALE metrics.

Proved non-existence of ALE K"ahler metrics with certain Ricci curvature signs on specific line bundles.

Abstract

We solve the geodesic equation in the space of K\"ahler metrics under the setting of asymptotically locally Euclidean (ALE) K\"ahler manifolds and we prove global $\mathcal{C}^{1,1}$ regularity of the solution. Then, we relate the solution of the geodesic equation to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of $\varepsilon$-geodesics at spatial infinity. Under the assumption that the Ricci curvature of a reference ALE K\"ahler metric is non-positive, convexity of the Mabuchi $K$-energy along $\varepsilon$-geodesics. However, we will also prove that on the line bundle $\mathcal{O}(-k)$ over $\mathbb{C}\mathbb{P}^{n-1}$ with $n \geq 2$ and $k \neq n$, no ALE K\"ahler metric can have non-positive (or non-negative) Ricci curvature.