Real Hypersurfaces, Orders of Contact, and Applications

A fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the Cauchy-Riemann equations. Suppose that D is an open domain in Cn, and that its boundary, M, is a smooth real submanifold of Cn. How does the geometry of M influence the function theory on D? One approach to this question is the a-Neumann problem, [6]. There one studies the inhomogeneous Cauchy-Riemann equations au -a where a is a (0, q) form with distribution coefficients on the closed domain D U M satisfying the necessary compatibility condition a a 0. The a-Neumann problem constructs a particular solution u which is orthogonal to the null space of the complex Laplacian. Kohn [10a] has solved this problem and has shown that good local regularity properties for u in terms of a follow from so-called subelliptic estimates. It is natural to seek geometric conditions on M that are necessary and sufficient for these estimates. In this paper we will be concerned with a certain geometric condition on M that we feel is the right one for subellipticity. Although the motivation for the questions discussed here comes from partial differential equations, the techniques come from algebraic geometry. To see why, we state theorems of Kohn, Greiner, and Catlin concerning subellipticity. Kohn [1Oa] has discovered a sufficient condition for a subelliptic estimate near a fixed point p in M. In case D is pseudoconvex and is contained in C2, this condition is equivalent to the finiteness of order of tangency of every complex analytic (one dimensional) manifold with M at p. Greiner [13] established the necessity of this condition in