Nahm’s equations and free boundary problems

In [4], following up work of Hitchin [9], the author found it useful to express Nahm’s equations, for a matrix group, in terms of the motion of a particle in a Riemannian symmetric space, subject to a potential field. This point of view lead readily to an elementary existence theorem for solutions of Nahm’s equation, corresponding to particle paths with prescribed end points. The original motivation for this article is the question of formulating an analogous theory for the Nahm equations associated to the infinite-dimensional Lie group of areapreserving diffeomorphisms of a surface–in the spirit of [5]. We will see that this can be done, and that a form of the appropriate existence theorem holds— essentially a special case of a result of Chen. However the main focus of the article is not on existence proofs but on the various formulations of the problem, and connections between them. In these developments, one finds that the natural context is rather more general than the original question, so we will start out of a different tack, and return to Nahm’s equations in Section 5. Consider the following set-up in Euclidean space R, in which we take coordinates (x1, x2, z)—thinking of z as the vertical direction. (We will use the notation ∂ ∂xi = ∂i, ∂ ∂z = ∂z .) Suppose we have a strictly positive function H(x1, x2). This defines a domain ΩH = {(x1, x2, z) : 0 < z < H(x1, x2)},