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The uniqueness of Poincaré type extremal Kähler metric

Yulun Xu

TL;DR

The paper extends the classical uniqueness results for extremal Kähler metrics to complete noncompact settings by analyzing Poincaré type metrics with cusp singularities along a divisor. Central to the method is a detailed solvability theory for the Lichnerowicz operator on weighted Hölder spaces, together with a careful decomposition of potentials into invariant and noninvariant parts and a robust control of the isometry group near and away from the divisor. Through a gauge-fixing argument and the study of extremal vector fields via the Futaki character, the authors prove that two Poincaré type extremal Kähler metrics in the same cohomology class are related by an automorphism in Aut_0^D(X) when Aut_0(D)=\{Id\}. This sharpens our understanding of uniqueness beyond closed manifolds and supports the broader program linking canonical metrics to algebraic stability in the noncompact cusp regime.

Abstract

Let $D$ be a smooth divisor on a closed Kähler manifold $X$. Suppose that $Aut_0(D)=\{Id\}$. We prove that the Poincaré type extremal Kähler metric with a cusp singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$. This generalizes Berman-Berndtson's work on the uniqueness of extremal Kähler metrics from closed manifolds to some complete and noncompact manifolds.

The uniqueness of Poincaré type extremal Kähler metric

TL;DR

The paper extends the classical uniqueness results for extremal Kähler metrics to complete noncompact settings by analyzing Poincaré type metrics with cusp singularities along a divisor. Central to the method is a detailed solvability theory for the Lichnerowicz operator on weighted Hölder spaces, together with a careful decomposition of potentials into invariant and noninvariant parts and a robust control of the isometry group near and away from the divisor. Through a gauge-fixing argument and the study of extremal vector fields via the Futaki character, the authors prove that two Poincaré type extremal Kähler metrics in the same cohomology class are related by an automorphism in Aut_0^D(X) when Aut_0(D)=\{Id\}. This sharpens our understanding of uniqueness beyond closed manifolds and supports the broader program linking canonical metrics to algebraic stability in the noncompact cusp regime.

Abstract

Let be a smooth divisor on a closed Kähler manifold . Suppose that . We prove that the Poincaré type extremal Kähler metric with a cusp singularity at is unique up to a holomorphic transformation on that preserves . This generalizes Berman-Berndtson's work on the uniqueness of extremal Kähler metrics from closed manifolds to some complete and noncompact manifolds.

Paper Structure

This paper contains 35 sections, 76 theorems, 354 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. Background metric of Poincaré type
  5. Asymptotic behaviour of Poincaré type extremal Kähler metrics
  6. Quasi coordinates
  7. Function spaces
  8. Poincaré type $C^{1,1}$ geodesic
  9. Energy functionals
  10. Fiber bundle structure of a neighbourhood of D
  11. Holomorphic vector fields
  12. Solvability of Lichnerowicz operator
  13. Kernel and range
  14. Kernel and holomorphic vector fields.
  15. u0 and u
  16. ...and 20 more sections

Key Result

Theorem 1.1

Suppose that $D$ is a smooth divisor on $X$ and $Aut_0(D)=\{Id\}$. Given two Poincaré type extremal Kähler metrics $\omega_1$ and $\omega_2$ in a given cohomology class $[\omega]$. Then there exists an element $g\in Aut_0^D(X)$ such that $g^* \omega_1 = \omega_2$.

Theorems & Definitions (137)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Proposition 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1
  • Lemma 3.2
  • ...and 127 more