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.