  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.
