MiS Preprint Repository

In this paper, we discuss the extremal Hermitian metrics with finite energy and area on compact Riemann surfaces with conical singularities. We obtain the following classification theorem of such metrics, which generalizes results of X. X. Chen: Let M be a compact Riemann surface, g an extremal Hermitian metric with finite energy and area on $M\mathrm{\setminus }\{p_j{\}}_j=1,...,n$ and K the Gaussian curvature of g. Then g is a conical metric with singular angles ${\alpha }_j(j=1,...,n)$ (which may include some weak cusps). Furthermore if all singular angles satisfy
$2\pi {\alpha }_j\le \pi$
then the following classifications hold:

1. If $genu \hbar (M) \geq 1$, then $K \equiv const.$;
2. If $M=S^2$ and $n \geq 3$, then $K \equiv const.$;
3. If $M=S^2$ and n=2, then there are two cases:
   1. if both singular points are cusp, then there is no extremal Hermitian metric;
   2. if one of singular points is not cusp, then g is a rotationally symmetric extremal Hermitian metric determined uniquely by the total area and two angles $2 \pi \alpha$.
4. If $M=S^2$ and n=1, then g is a rotationally symmetric metric determined uniquely by the total area and angle $2 \pi \alpha$.