The area minimizing problem in conformal cones, II

TL;DR

This paper investigates the connection between area minimizing problems, area functionals, and minimal surface equations in conformal cones, extending classical results from hyperbolic spaces to more general conformal manifolds.

Contribution

It generalizes the structure of area minimizing currents and solves Dirichlet problems for minimal surfaces in conformal cones, broadening the scope beyond hyperbolic spaces.

Findings

Describes the structure of area minimizing currents in conformal cones.

Solves Dirichlet problems for minimal surface equations under mean convexity.

Extends existence and uniqueness results to a wider class of conformal manifolds.

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 \cite{GZ20}. These cones are certain generalizations of hyperbolic spaces. We describe the structure of area minimizing $n$-nteger multiplicity currents in bounded $C^2$ conformal cones with prescribed $C^1$ 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.