The existence of solutions to a $k$-sum equation arising from conformal geometry

Abstract

On compact manifolds $(M,g)$, we derive the existence of metrics in a given conformal class $[g]$ with prescribed negative partial curvature. This curvature corresponds to a fully nonlinear equation derived from conformal geometry. For manifolds with boundary, we demonstrate the solvability of equations involving prescribed negative partial curvature within $M$, coupled with mean curvature along $\partial M$.