Prescribed Gauss curvature problem on singular surfaces

Abstract

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $$\Sigma $$ admitting conical singularities of orders $$\alpha _i$$ ’s at points $$p_i$$ ’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $$\chi (\Sigma )+\sum _i \alpha _i$$ approaches a positive even integer, where $$\chi (\Sigma )$$ is the Euler characteristic of the surface $$\Sigma $$ .