Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of ${\bf C}\sp{n}$

Abstract

The relationship between the Levi geometry of a submanifold of C' and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of Cn, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in Cn. This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of Cn. In fact, we show that if S is a real analytic, generic, submanifold of C' (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then S is not extendible to any open set in C. 0. Introduction. Recently, there has been increasing interest in the relationship of geometric concepts, such as the Levi form of a hypersurface and the local solvability of the tangential Cauchy-Riemann equations. For example, Henkin [H] has shown that on a strictly pseudoconvex hypersurface, one can locally solve the tangential Cauchy-Riemann equations (in most bidegrees). His approach involved explicit integral kernels. In this paper we start such a program for higher codimension. We relate such concepts as the Levi form and the Levi algebra to the local solvability of the tangential Cauchy-Riemann equations on a real analytic, codimension two submanifold of Cn. The geometric conditions we impose are in terms of strict pseudoconvexity of the Levi form (Definition 2.3) and the excess dimension of the Levi algebra (Definition 2.2). We represent the local solution to the tangential Cauchy-Riemann equations in terms of explicit kernels. We show (Theorem 5.1) that on a generic real analytic, strictly pseudoconvex, codimension two submanifold S of Cn with excess dimension of S equal to one, we can locally solve the tangential Cauchy-Riemann equations in bidegrees (p, q) where 1 s q < n 3. Although our emphasis is on codimension two, we indicate an inductive process for local solvability on higher codimension submanifolds of Cn. From the kernel approach we also get, for free, a local result (Theorem 5.15) about CR-functions on a codimension two submanifold, S, satisfying the hypothesis of Theorem 5.1. Locally, we carefully construct an open set V in C n with V n s F 0 such that a CR-function on V n s is the boundary value jump across S of a Received by the editors January 24, 1980 and, in revised form, February 9, 1981. 1980 Mathematics Subject Classification. Primary 35N15, 32F99. (DI 982 American Mathematical Society 0002-9939/82/00001064/$06.75