Robin and Neumann problems for the graph scalar curvature equation

Abstract

We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumption, we prove existence and uniqueness for the Robin problem and obtain a classical Neumann solution as the Robin parameter tends to zero. The volume threshold is optimal among conditions depending only on the volume. The main step is a boundary second-derivative estimate uniform in the Robin parameter; known interior and global-to-boundary curvature estimates then give the global bound.