Contents Online
Mathematical Research Letters
Volume 30 (2023)
Number 2
Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics
Pages: 541 – 576
DOI: https://dx.doi.org/10.4310/MRL.2023.v30.n2.a8
Author
Abstract
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 [$\href{https://doi.org/10.1112/plms.12255}{31}]$. This extends a result by Berman–Berndtsson [$\href{https://doi.org/10.1090/jams/880}{7}$] and Chen–Paun–Zeng [$\href{https://arxiv.org/pdf/1506.01290.pdf}{17}$] 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 [$\href{https://doi.org/10.1112/plms.12255}{31}$] 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