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

TL;DR

This paper explores real projective connections on Riemann surfaces and their Liouville equation solutions, revealing black hole type singularities on contours, with detailed analysis of the four-punctured sphere case.

Contribution

It connects Smirnov's approach to projective connections with black hole type solutions of the Liouville equation on Riemann surfaces.

Findings

Solutions exhibit singularities of black hole type on specific contours.

Detailed analysis of the four-punctured Riemann sphere case.

Identification of special singularity structures in the solutions.

Abstract

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.