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

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 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 ${\mathbb{C}}^n$ 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 ${\mathbb{R}}^3$.