The area minimizing problem in conformal cones, I

摘要

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Euclidean 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 minimizing problems. We can solve the area minimizing problem in nonmean 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. (C) 2020 Elsevier Inc. All rights reserved.