The Heat Equation Method of Milgram and Rosenbloom for Open Riemannian Manifolds

Abstract

In a pair of notes [12] in the Proceedings of the National Academy of Sciences, Milgram and Rosenbloom introduced a generalized heat equation into the theory of harmonic integrals and used it to obtain a proof of the existence assertion of Hodge's theorem. The theory of integral equations of Volterra type played an important role in the presentation, which was restricted to compact manifolds. This restriction to compact manifolds was eased by Spencer [15], who extended the methods to apply to a class of complete manifolds in which strong restrictions are put on the curvature tensor. A generalization in a different direction was made by Yosida [18], who replaced the integral equation methods by Hilbert space methods and obtained results for open manifolds. Spencer [17] used these methods in examining Green's and Neumann's boundary value problems for A. Lax and Milgram have used Hilbert space techniques in applying the heat equation method to the study of certain partial differential equations. Yosida showed that the equation (a at)at = Aat can be solved in such a way that if a (with finite norm) is the initial value of at, then a. is in the space of those norm-finite forms 3 for which Ai = 0. (We take the sign of A to be such that for functions A reduces to the negative of the Laplace-Beltrami operator.) He described his results as an ergodic theorem in which the solution of the diffusion equation leads to a stochastic procedure for going from a form to its harmonic part. For closed manifolds he obtained additional information adequate to demonstrate Hodge's existence theorem. The methods used in this paper are patterned after those of Yosida but lead to a more complete generalization of the results of Milgram and Rosenbloom. Denoting the solution of the heat equation with initial value a by Wta, the main results are (complete hypotheses below): W. is the projection onto the harmonic forms (those forms with finite norm for which both da and ba are zero). If a is closed, Wta is closed. If a is closed, the periods of Wta on compact cycles are independent of t, 0 < t < cc? Thus W. projects a onto a harmonic form with the same periods.