On real projective connections, V.I. Smirnov's approach, and black hole type solutions of the Liouville equation

We consider real projective connections on Riemann surfaces and corresponding solutions of the Liouville equation. It is shown that these solutions have singularities of special type (of a black hole type) on a finite number of simple analytical contours. The case of the Riemann sphere with four real punctures, considered in V.I. Smirnov’s thesis (Petrograd, 1918), is analyzed in detail.