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Weighted cscK metric (II): the continuity method

Eleonora Di Nezza, Simon Jubert, Abdellah Lahdili

TL;DR

This work develops a comprehensive framework for metrics with weighted constant scalar curvature on compact Kähler manifolds endowed with a torus action. By introducing the $v$-weighted scalar curvature and the $(v,w)$-cscK equation, the authors prove that coercivity of the weighted Mabuchi energy $oldsymbol{M}_{v,w}$ implies the existence of a $(v,w)$-cscK metric, establishing a precise equivalence and unifying several geometric notions (classical cscK, Calabi extremal, weighted solitons, and extremal metrics on semisimple fibrations). The methodology hinges on a weighted Chen continuity path, with detailed openness, closeness, and regularity results, including robust $C^0$, $W^{1,2p}$, and entropy-based estimates. The paper also develops a toric Yau–Tian–Donaldson correspondence in the weighted setting and extends these results to semisimple principal fibrations, providing practical stability criteria and sufficient conditions in the toric case. Collectively, the findings offer powerful tools for analyzing weighted cscK metrics and their geometric consequences in toric and fibred contexts, with potential impact on stability notions and moduli.

Abstract

In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact Kähler manifold $X$: this notion include constant scalar curvature Kähler metrics, weighted solitons, Calabi's extremal Kähler metrics and extremal metric on semisimple principal fibrations. We prove that the coercivity of the weighted Mabuchi functional implies the existence of a wcscK metric, thereby achieving the equivalence. \\ We then give several applications in Kähler and toric geometry, such as a weighted version of the toric Yau-Tian-Donaldson correspondence, and the characterization of the existence of wcscK metric on total space of semisimple principal fibration $Y$ in term of existence of wcscK metric on its fiber $X$.

Weighted cscK metric (II): the continuity method

TL;DR

This work develops a comprehensive framework for metrics with weighted constant scalar curvature on compact Kähler manifolds endowed with a torus action. By introducing the -weighted scalar curvature and the -cscK equation, the authors prove that coercivity of the weighted Mabuchi energy implies the existence of a -cscK metric, establishing a precise equivalence and unifying several geometric notions (classical cscK, Calabi extremal, weighted solitons, and extremal metrics on semisimple fibrations). The methodology hinges on a weighted Chen continuity path, with detailed openness, closeness, and regularity results, including robust , , and entropy-based estimates. The paper also develops a toric Yau–Tian–Donaldson correspondence in the weighted setting and extends these results to semisimple principal fibrations, providing practical stability criteria and sufficient conditions in the toric case. Collectively, the findings offer powerful tools for analyzing weighted cscK metrics and their geometric consequences in toric and fibred contexts, with potential impact on stability notions and moduli.

Abstract

In this paper we investigate the existence of metrics with weighted constant scalar curvature (wcscK for short) on a compact Kähler manifold : this notion include constant scalar curvature Kähler metrics, weighted solitons, Calabi's extremal Kähler metrics and extremal metric on semisimple principal fibrations. We prove that the coercivity of the weighted Mabuchi functional implies the existence of a wcscK metric, thereby achieving the equivalence. \\ We then give several applications in Kähler and toric geometry, such as a weighted version of the toric Yau-Tian-Donaldson correspondence, and the characterization of the existence of wcscK metric on total space of semisimple principal fibration in term of existence of wcscK metric on its fiber .

Paper Structure

This paper contains 23 sections, 45 theorems, 230 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Estimates for a weighted cscK type equation
  4. Estimates for uniform norm
  5. Integral second order estimates
  6. Integral estimates for the gradient
  7. Openness and existence of a twisted solution
  8. The weighted continuity path of Chen
  9. Linearization of the continuity path
  10. Existence of a solution for $t$ small
  11. Construction of an approximate solution
  12. Applying the implicit function theorem
  13. Closeness
  14. Weighted Aubin-Mabuchi functionals
  15. Closeness for time less than one
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathrm{v}>0$, $\mathrm{w}$ be two weight functions on $P$ such that $\mathrm{v}$ is $\log$-concave and $\mathrm{v},\mathrm{w}$ satisfy for any $\varphi\in \mathcal{K}(X,\omega_0)^{\mathbb{T}}$. Assume that the weighted Mabuchi energy $\mathbf {M}_{\mathrm{v},\mathrm{w}}$ is coercive, i.e. there exist positive constants $A, B\in \mathbb{R}$ such that for any $\varphi\in \mathcal{K}(X,\omega

Figures (1)

  • Figure 1: The cone decomposition

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • ...and 64 more