Sidney M. Webster

Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations

It is well known that the complex analytical properties of a real submanifold M in the complex space C n are most accessible through consideration of the complex tangents to M. The properties we have in mind are related to the behavior of holomorphic functions on or near M and to the behavior of M under biholomorphic transformation. The case in which M is a real hypersurface is most familiar, while much less is known for higher codimension. In this paper we consider the critical case of a real ndimensional manifold M in C n, which we also assume to be real analytic. At a generic point M is locally equivalent to the standard R n in C n. However, near a complex tangent M may aquire a non-trivial local hull of holomorphy and other biholomorphic invariants. We begin with the simplest non-trivial case, which is a surface M2cC 2 with an isolated, suitably non-degenerate complex tangent. Here one already encounters a rich structure and non-trivial problems. In coordinates zj=xi+iy i, j= 1,2, M may be written locally as

On the proof of Kuranishi's embedding theorem

Abstract We prove a local holomorphic embedding theorem for a formally integrable, strictly pseudoconvex CR manifold M with dim M = 2n − 1 ≧ 7. This embedding is obtained as the limit of a sequence of approximate embeddings into complex n-space, which is constructed and shown to converge by the methods of Nash and Moser. The linearized problem is solved using the explicit integral operators constructed by Henkin. With estimates wich we have previously obtained for these operators, we show that if M is of class Cm, then it admits a Ck embedding provided 21 ≦ k, 6k + 5n − 2 ≦ m. Our argument is much shorter and simpler than previous arguments, which were based on the Neumann operator and carried out in the C∞ category.

On the local solution of the tangential Cauchy-Riemann equations

Abstract We study the solution operators P and homotopy formula introduced by G. M. Henkin for the tangential Cauchy-Riemann complex of a suitable small domain D on a strictly pseudoconvex real hypersurface in complex n-space. The main difficulties stem from the fact that P is an integral operator with a rather complicated kernel. For U ⊂⊂ D, we derive a Ck-norm estimate of the form ∥Pφ∥U, k ≦ K∥φ∥D, k, where the constant K blows up as U increases to D. We obtain careful control of the rate of this blow-up and of the dependence of K on the derivatives of the function defining the real hypersurface. Our estimates are sufficient for application to the local CR embedding problem.

Geometric and Dynamical Aspects of Real Submanifolds of Complex Space

This work is founded on an analogy between complex analysis and classical mechanics, which at first glance may not seem too meaningful. Our purpose is to show, however, that not only is it useful as a formal guide, but that the interplay is substantial at the level of mathematical proof.