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
**R**^{n}), 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 b^{2}
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).