A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions

Abstract

We develop a conformal duality for space‐like graphs in Riemannian and Lorentzian three‐manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is time‐like in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz–Minkowski space with prescribed mean curvature a bounded function with bounded gradient. On the other hand, we obtain conditions for the existence and nonexistence of entire graphs which are related to a notion of the critical mean curvature.