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) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msup> </m:math> {C^{1,\alpha}} fixed graphical boundaries, under a barrier condition. This result provides a generalization of 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {C^{1}} graphs.