The area minimizing problem in conformal cones, II
Abstract
In this paper we continue to study the connection among the area minimizing problem, certain area functional and the Dirichlet problem of minimal surface equations in a class of conformal cones with a similar motivation from GZ20. These cones are certain generalizations of hyperbolic spaces. We describe the structure of area minimizing n-nteger multiplicity currents in bounded C2 conformal cones with prescribed C1 graphical boundary via a minimizing problem of these area functionals. As an application we solve the corresponding Dirichlet problem of minimal surface equations under a mean convex type assumption. We also extend the existence and uniqueness of a local area minimizing integer multiplicity current with star-shaped infinity boundary in hyperbolic spaces into a large class of complete conformal manifolds.