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In the 1980s, Eugenio Calabi introduced the concept of {\it extremal K\"ahler metrics} as critical points of the $L^2$-norm functional of scalarcurvature in the space of K\" ahler metrics belonging to a fixed K\"ahler classof a compact complex manifold $X$. Calabi demonstrated that extremal K\" ahlermetrics always degenerate into Einstein metrics on compact Riemann surfaces. Wedefine a K\"ahler metric $g$ on a domain of ${\Bbb C}^n$ as a {\it localextremal K\"ahler metric} of dimension $n$ if it satisfies the Euler-Lagrangeequation of this functional, i.e. holomorphic is the $(1,0)$-part of thegradient vector field of the scalar curvature of $g$, in the domain. Our mainresult establishes that the space of all germs of local extremal, non-EinsteinK\"ahler metrics of dimension one comprises three components, eachdiffeomorphic to ${\Bbb R}^3$.

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