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Metrics of constant curvature 1 with three conical singularities on the 2-sphere. (English) Zbl 0958.30029
Summary: Let \(\text{Met}_1(\Sigma)\) be the set of positive semi-definite conformal metrics of constant curvature 1 with conical singularities on a compact Riemann surface \(\Sigma\). Suppose that \(d\sigma^2 \in\text{Met}_1 (\Sigma)\) has conical singularities at points \(p_j\in\Sigma (j=1,\dots,n)\) with order \(\beta_j(>-1)\), that is, it admits a tangent cone of angle \(2\pi(\beta_j +1)\) at each \(p_j\). A formal sum \(D=\sum^n_{j=1} \beta_jp_j\) is called the divisor of \(d\sigma^2\). Then the Gauss-Bonnet formula implies that \(\chi (\Sigma, D):= \chi(\Sigma): =\chi(\Sigma) +\sum^n_{j=1} \beta_j>0\). The divisor \(D\) is called subcritical, critical, or supercritical when \(\delta(\Sigma,D): = \chi (\Sigma,D) -2\text{Min}_{j=1, \dots,n} \{1,\beta_j +1\}\) is negative, zero, or positive, respectively. M. Troyanov [Trans. Am. Math. Soc. 324, No. 2, 793-821 (1991; Zbl 0724.53023)] showed that if \(\chi(\Sigma,D)>0\), there exists a pseudometric in \(\text{Met}_1(\Sigma)\) with divisor \(D\) whenever it is subcritical. On the other hand, for the supercritical case several obstructions are known and the existence problem of the metrics is difficult: M. Troyanov [Lect. Notes Math. 1410, 296-306 (1989; Zbl 0697.53037) gave a classification of metrics of constant curvature 1 with at most two conical singularities on the 2-sphere. In the paper, the authors gave a necessary and sufficient condition for the existence and uniqueness of a metric with three conical singularities of given order on the 2-sphere. As shown by the authors, there is a one to one correspondence between the set \(\text{Met}_1(\Sigma)\) and the set of branched CMC-1 (constant mean curvature one) immersions of \(\Sigma\) excluded finite points into the hyperbolic 3-space with given hyperholic Gauss map. To show the theorem, this correspondence plays an important role. It should be remarked that classical work of F. Klein (Vorlesungen über die hypergeometrische Funktion (1933; Zbl 0461.33001) is related to the paper.

30F10 Compact Riemann surfaces and uniformization
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature