The space of germs of extremal K\" ahler metrics in one dimension comprises three distinct ${\Bbb R}^3$ components

Abstract

In the 1980s, Eugenio Calabi introduced the concept of {\it extremal K\" ahler metrics} as critical points of the $L^2$-norm functional of scalar curvature in the space of K\" ahler metrics belonging to a fixed K\"ahler 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 K\"ahler metric $g$ on a domain of ${\Bbb C}^n$ as a {\it local extremal K\"ahler 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 K\"ahler metrics of dimension one comprises three components, each diffeomorphic to ${\Bbb R}^3$.