Prescribed mean curvature hypersurfaces in conformal product manifolds

Abstract

In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes Gerhardt's work to non-flat conformal backgrounds. As a consequence, we obtain new solutions to the high-dimensional PMC Plateau problem with explicitly characterized topology. Moreover, under a quasi-decreasing condition on the PMC function, we demonstrate that the resulting hypersurfaces are $C^1$ graphs.