The area minimizing problem in conformal cones

TL;DR

This paper investigates the existence of minimal graphs in conformal cones, generalizing classical geometric problems, and establishes conditions under which area minimizing solutions exist in various conformal cone settings.

Contribution

It introduces the NCM condition for bounded domains and proves the existence of minimal graphs in mean convex conformal cones, extending classical minimal surface theory.

Findings

Existence of bounded minimal graphs in mean convex conformal cones.

Minimal graphs solve the area minimizing problem under NCM condition.

Extension of results to non-mean convex cones contained in larger mean convex cones.

Abstract

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Eulcidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contained in a larger mean convex conformal cones with the NCM assumption. We give examples to illustrate that this assumption can not be removed for our main results.