I have a number of results on the existence of solutions to the Plateau problem for surfaces of prescribed mean curvature. One of the more interesting papers [8] holds for surfaces in higher codimension, and introduces the natural concept of a surface with prescribed mean curvature vector. Given an alternating (2,1)-tensor H on a Riemannian manifold (such as Rn), a surface is said to have prescribed mean curvature vector H if at each point x, the mean curvature vector of the surface equals the value of H on an orthonormal basis of the tangent plane to the surface. Choose a point p in the manifold, and let b2 be an upper bound for sectional curvatures. One result is that if H is formed by raising an index on an exact 3-form and has radial component at most b cot(br) when evaluated on any orthonormal pair, where r is the distance to p, then any Jordan curve in the ball of radius r centered at p is the boundary of a branched disk of prescribed mean curvature vector H. These bounds are sharp [19]. Analogous theorems for currents of dimension greater than 2 have been proved by Klaus Steffen and Frank Duzaar.

[8] Existence of Surfaces with Prescribed Mean Curvature Vector, Math. Z. 131, 117-140 (1973).
[19] Necessary Conditions for Submanifolds and Currents with Prescribed Mean Curvature Vector, Annals of Mathematics Studies 103, 225-242 (1983).