Mathematical Research Letters

Volume 30 (2023)

Number 2

Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics

Pages: 541 – 576



Abdellah Lahdili (Department of Mathematics, University of Aarhus, Denmark)


We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kähler metrics on a compact Kähler manifold introduced in our previous work [$\href{}{31}]$. This extends a result by Berman–Berndtsson [$\href{}{7}$] and Chen–Paun–Zeng [$\href{}{17}$] in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy, thus extending our results from [$\href{}{31}$] from the polarized to the Kähler case. This implies a suitable notion of weighted $K$-semistability of a Kähler manifold admitting a weighted extremal Kähler metric.

Received 2 July 2020

Received revised 16 December 2022

Accepted 7 February 2023

Published 13 September 2023