Prescribed mean curvature hypersurfaces in conformal product manifolds

2024/01/25 by Gao, Qiang, Zhou, Hengyu
#49Q20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2401.14097

Abstract

In this paper we give the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) C^1,α fixed graphical boundaries under a barrier condition. This generalizes Gerhardt's result in conformally flat spaces. It provides new examples of the Plateau problem of PMC hypersurfaces with clear topology under high dimensions. In addition, if a quasi-decreasing condition of PMC functions is satisfied, such PMC hypersurfaces are C¹ graphs.

Related

• Conformal Killing graphs with prescribed mean curvature
• Closed hypersurfaces of prescribed mean curvature in locally conformally flat Riemannian manifolds
• Prescribed mean curvature surfaces in the product spaces \mathbbM²(κ)×ℝ; Height estimates and classification results for properly embedded surfaces
• Geometric barriers for the existence of hypersurfaces with prescribed curvatures in Mⁿ× R
• A blow-up method to prescribed mean curvature graphs with fixed boundaries