The geometric Neumann problem for the Liouville equation

TL;DR

This paper classifies solutions to the geometric Neumann problem for the Liouville equation in half-plane domains, characterizing conformal metrics with finite area, boundary singularities, and constant geodesic curvature.

Contribution

It provides a complete classification of solutions with finite area and boundary singularities for the Liouville equation under Neumann boundary conditions.

Findings

Classified solutions with finite area and boundary singularities.

Characterized conformal metrics with constant curvature and boundary conditions.

Identified solutions not restricted to conical singularities.

Abstract

In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.