The space of germs of extremal K" ahler metrics in one dimension comprises three distinct \Bbb R³ components

2023/11/27 by Chen, Qing, Shi, Yiqian, Xu, Bin
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 58E11 #Secondary 53B35

paper · doi:10.48550/arxiv.2311.15461

Abstract

In the 1980s, Eugenio Calabi introduced the concept of \it extremal K" ahler metrics as critical points of the L²-norm functional of scalar curvature in the space of K" ahler metrics belonging to a fixed Kähler class of a compact complex manifold X. Calabi demonstrated that extremal K" ahler metrics always degenerate into Einstein metrics on compact Riemann surfaces. We define a Kähler metric g on a domain of \Bbb Cⁿ as a \it local extremal Kähler metric of dimension n if it satisfies the Euler-Lagrange equation of this functional, i.e. holomorphic is the (1,0)-part of the gradient vector field of the scalar curvature of g, in the domain. Our main result establishes that the space of all germs of local extremal, non-Einstein Kähler metrics of dimension one comprises three components, each diffeomorphic to \Bbb R³.

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