On the Existence of Weighted-cscK Metrics

TL;DR

This paper establishes that on smooth Kähler manifolds, the coercivity of a weighted Mabuchi functional guarantees the existence of various weighted constant scalar curvature Kähler metrics, including cscK and Kähler-Ricci solitons.

Contribution

It proves a new link between the coercivity of a weighted Mabuchi functional and the existence of weighted cscK metrics, extending previous studies to a broader class of metrics.

Findings

Weighted Mabuchi functional coercivity implies existence of weighted cscK metrics.

Includes existence results for Kähler-Ricci solitons and μ-cscK metrics.

Generalizes previous results to log-concave weight functions.

Abstract

In this paper, we prove that on a smooth K\"ahler 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, K\"ahler-Ricci solitons, $\mu$-cscK metrics.