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Geodesic Equations on asymptotically locally Euclidean Kähler manifolds

Qi Yao

Abstract

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.

Geodesic Equations on asymptotically locally Euclidean Kähler manifolds

Abstract

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 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 -geodesics at spatial infinity. Under the assumption that the Ricci curvature of a reference ALE Kähler metric is non-positive, convexity of the Mabuchi -energy along -geodesics. However, we will also prove that on the line bundle over with and , no ALE Kähler metric can have non-positive (or non-negative) Ricci curvature.
Paper Structure (8 sections, 18 theorems, 175 equations)

This paper contains 8 sections, 18 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. $\varepsilon$-geodesic equations and openness
  3. A priori estimate up to $\mathcal{C}^0$
  4. A priori estimate up to $\mathcal{C}^1$
  5. A priori estimate up to $\mathcal{C}^{1,1}$
  6. The asymptotic behavior of $\varepsilon$-geodesics
  7. Convexity of the Mabuchi $K$-energy
  8. Nonexistence of non-positive (or non-negative) Ricci curvature

Key Result

Theorem 1.1

(Yao 2022) Let $(X, J)$ be an ALE Kähler manifold asymptotic to $\mathbb{C}^n / \Gamma$. Let $\omega_1$, $\omega_2$ be Kähler forms in the same Kähler class of $(X, J)$ with the corresponding metrics satisfying decayale and such that the scalar curvatures of $\omega_1$ and $\omega_2$ are equal, $R_1 for some $\tilde{\tau} >n-1$ depending on $(n, \tau)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 20 more