Real Projective Connections, v. I. Smirnov’s Approach, and Black-hole-type Solutions of the Liouville Equation
نویسنده
چکیده
One of the central problems of mathematics in the second half of the 19th century and at the beginning of the 20th century was the problem of uniformization of Riemann surfaces. The classics, Klein [1] and Poincaré [2], associated it with studying second-order ordinary differential equations with regular singular points. Poincaré proposed another approach to the uniformization problem [3]. It consists in finding a complete conformal metric of constant negative curvature, and it reduces to the global solvability of the Liouville equation, a special nonlinear partial differential equation of elliptic type on a Riemann surface. Here, we illustrate the relation between these two approaches and describe solutions of the Liouville equation corresponding to second-order ordinary differential equations with a real monodromy group. In the modern physics literature on the Liouville equation, it is rather commonly assumed that for the Fuchsian uniformization of a Riemann surface, it suffices to have a second-order ordinary differential equation with a real monodromy group. But the classics already knew that this is not the case, and they analyzed second-order ordinary differential equations with a real monodromy group on genus-0 Riemann surfaces with punctures in detail. Nonetheless, they did not consider the relation to the Liouville equation, and we partially fill this gap here. Namely, in Sec. 2, following the lectures [4], we briefly describe the theory of projective connections on a Riemann surface—an invariant method for defining a corresponding second-order ordinary differential equation with regular singular points. Following [5], [6], we review the main results on the Fuchsian uniformization, the Liouville equation, and the complex geometry of the moduli space. In Sec. 3, following [7], we present the modern classification of projective connections with a real monodromy group and review the results of V. I. Smirnov’s thesis [8] (Petrograd, 1918). This work, published in [9], [10], was the first
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