Prescribed combinatorial curvature beyond positivity for spherical vertex scalings on the sphere

Abstract

Let $(S,V)$ be a marked 2-sphere with $|V|\ge3$, and fix a spherical discrete conformal class. Izmestiev-Prosanov-Wu proved existence and uniqueness for positive prescribed combinatorial curvature in the spherical discrete conformal class on $(S,V)$. In this paper, we identify a region of prescribed combinatorial curvature vectors, allowing some negative components, for which existence still holds.