A Dirichlet Principle for Analytic Functions of Several Complex Variables

Abstract

The theory of analytic varieties of several complex dimensions (more than one) as distinguished from the theory of Riemann surfaces, contains no general theorems of classification and existence. It is known that the direct generalizations of Riemann's theorems concerning the existence of non-trivial meromorphic functions and Abelian differentials are false for the several variable case. Further the identification of compact Riemann surfaces with algebraic curves has no analogue in several variables. Riemann's original account of his theorems was founded upon a variational method, the celebrated Dirichlet Principle. In this paper we shall generalize this variational method, that is, we formulate a variational problem whose extremals are analytic functions of several complex variables. We give a necessary and sufficient condition for the existence of a non-trivial bounded complex analytic density on an arbitrary complex analytic manifold. The restriction to boundedness (which is defined later on) is superfluous for compact manifolds. Our argument makes no distinction between the one variable and several variable case. The formulation of our result is given in terms of Hilbert spaces as suggested by Weyl's paper on the method of orthogonal projections. We associate with an arbitrary complex analytic manifold V, two Hilbert spaces V3* and 3*. 3* is a closed subspace of .3** The space 8* is given by an explicit construction, while 23* is defined indirectly. The main theorem asserts that the orthogonal complement of S* in V3* is precisely the space of bounded comlpex analytic densities on V. There is a vacuous element to this result; that is, nothing is asserted concerning the proper inclusion of 3* in p3*. This was to have been expected as an arbitrary complex manifold does not necessarily have any non-trivial complex analytic densities. For the case of Riemann surfaces one settles the question of the existence of non-trivial Abelian integrals of the first kind with the aid of topological considerations. There are no known topological conditions, applicable to arbitrary higher dimensional manifolds, that would assure the existence of non-trivial complex analytic densities. 1. V is a complex analytic manifold of n complex dimensions. DEFINITION 1. Suppose we have assigned to each pair (p, (ui, . , u,)) consisting of a point p of V and an ordered set u1, .. *, u of complex coordinates of V centered at the point p, a complex valued function fp,(u) defined in some nbd. of p, in such a fashion that