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On the Existence of Weighted-cscK Metrics

Jiyuan Han, Yaxiong Liu

TL;DR

This work advances the theory of weighted-cscK metrics by proving that, on a compact Kähler manifold with a log-concave weight $v$, the existence of a unique $(v,w)$-weighted-cscK metric (modulo automorphisms) is equivalent to the ${\mathbb G}$-coercivity of the weighted Mabuchi functional ${\mathbb M}_{v,w\cdot\ell_{\mathrm{ext}}}$. The authors formulate the weighted-cscK problem as a coupled PDE system and develop a robust a priori estimate framework, combining a modified Guo–Phong $C^0$-estimate with Chen–Cheng-type $W^{2,p}$ and gradient controls under log-concavity. They implement a continuity path à la Chen–Cheng and Hasimoto–Cheng, proving openness via a careful linearization and closedness via compactness arguments ensured by coercivity. The results encompass classical canonical metrics (Kähler–Einstein, cscK, solitons, and $\mu$-cscK) within a unified variational setting, providing a potential algebraic criterion for solvability in the weighted-cscK category and enriching the Yau–Tian–Donaldson-type program in this generalized context.

Abstract

In this paper, we prove that on a smooth Kähler manifold, the $\mathbb{G}$-coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in \cite{Lah19}), e.g, cscK metrics, Kähler-Ricci solitons, $μ$-cscK metrics.

On the Existence of Weighted-cscK Metrics

TL;DR

This work advances the theory of weighted-cscK metrics by proving that, on a compact Kähler manifold with a log-concave weight , the existence of a unique -weighted-cscK metric (modulo automorphisms) is equivalent to the -coercivity of the weighted Mabuchi functional . The authors formulate the weighted-cscK problem as a coupled PDE system and develop a robust a priori estimate framework, combining a modified Guo–Phong -estimate with Chen–Cheng-type and gradient controls under log-concavity. They implement a continuity path à la Chen–Cheng and Hasimoto–Cheng, proving openness via a careful linearization and closedness via compactness arguments ensured by coercivity. The results encompass classical canonical metrics (Kähler–Einstein, cscK, solitons, and -cscK) within a unified variational setting, providing a potential algebraic criterion for solvability in the weighted-cscK category and enriching the Yau–Tian–Donaldson-type program in this generalized context.

Abstract

In this paper, we prove that on a smooth Kähler manifold, the -coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in \cite{Lah19}), e.g, cscK metrics, Kähler-Ricci solitons, -cscK metrics.
Paper Structure (15 sections, 24 theorems, 227 equations)

This paper contains 15 sections, 24 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weighted Monge-Ampère measures, space of potentials, and functionals
  4. Weighted-cscK equations
  5. Examples
  6. Apriori estimates
  7. Uniform -estimate
  8. -estimate
  9. Gradient estimate of
  10. Proof of Continuity Method
  11. Openness
  12. Closedness
  13. Appendix
  14. Apriori estimates for weighted cscK equation
  15. Linearization of operators

Key Result

Theorem 1.2

Assume $\mathrm v$ is log-concave. The following statements are equivalent: (1). Up to the automorphism group ${\rm Aut}_T(X)$, there exists a unique solution of the weighted-cscK (extremal) equation weighted-csck. (2). The weighted Mabuchi functional for weighted-cscK (extremal) equation is $\mathb for some constants $\delta>0$, $C>0$, and any $\phi\in{\mathcal{E}}^1_{K}(X, \omega)$.

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1: BWN14HL20
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • ...and 44 more