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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.01991 |
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Table of Contents:
- We solve the geodesic equation in the space of Kähler metrics under the setting of asymptotically locally Euclidean (ALE) Kähler 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 Kähler 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 Kähler metric can have non-positive (or non-negative) Ricci curvature.