C. Denson Hill

A METHOD FOR THE CONSTRUCTION OF REFLECTION LAWS FOR A PARABOLIC EQUATION(

with b2ac <0 whose coefficients are holomorphic functions of 6, Xj and r. We present a general method for the construction of explicit reflection formulae (analogous to the classical Schwarz reflection principle for harmonic functions) for solutions of (1.1) which vanish along a noncharacteristic analytic surface. These formulae (cf. (8.3) and (8.7)) have a domain of dependence consisting, in general, of a one-dimensional curve extending from the reflecting surface to a specific image point. In special cases, however, the domain of dependence may degenerate to just the image point. An interesting aspect of our technique is the use of fundamental solutions which have singularities not only along the real characteristic r = const. = t associated with (1.1) but also on certain complex characteristics as well. The simplest case of (1.1) is the heat equation

The complex Frobenius theorem for rough involutive structures

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on previous recent work of the authors.

What is the Notion of a Complex Manifold with a Smooth Boundary

Publisher Summary This chapter show how there are two distinct notions of what constitutes a complex manifold with a smooth (C∞) boundary. A concrete boundary consists of prescribing an atlas of holomorphic coordinate charts. An abstract boundary consists of prescribing an integrable almost complex structure that is C∞ up to the boundary. For a complex manifold without boundary, these two prescriptions are equivalent, according to the well known Newlander-Nirenberg theorem. But when a boundary is present, the chapter shows by examples that the two concepts do not coincide; the abstract boundary is the more general concept. One could imagine that in dealing with a complex manifold with a smooth boundary, the classical complex analyst has at his disposal two maximal atlases: the atlas of all C∞ charts, and the atlas of all holomorphic charts. The abstract boundary is really the intrinsic notion; the concrete boundary has some extrinsic flavor. The chapter also discusses the influence of real analyticity and the influence of pseudoconvexity.

On the nonvanishing of abstract Cauchy–Riemann cohomology groups

In this paper we prove infinite dimensionality of some local and global cohomology groups on abstract Cauchy–Riemann manifolds.

Conormal suspensions of differential complexes

§1 An exact sequence for Whitney functions. §2 A generalized Mayer–Vietoris sequence. §3 Carriers and wedge decomposition. §4 Localization. §5 Wedge decomposition for CR manifolds. §6 The de Rham complex associated to a regular assignment of closed sets. §7 The long exact sequence associated to a regular assignment of closed sets. §8 Differential equivalence. §9 Local isomorphisms for the ∆–complex. §10 Tangential complexes. §11 Conormal suspensions. §12 On the mixed de Rham–Dolbeault complex. §13 Poincare lemma and holomorphic extension for the tangential Cauchy–Riemann complex. §14 Solvability wave front sets. §15 Degree q cohomology for q–pseudoconcave CR manifolds. §16 Solvability wave front sets and wedge decomposition for q–pseudoconcave CR manifolds.

Non Completely Solvable Systems of Complex First Order PDE's

We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.s, especially related to the analysis on CR manifolds.

Holomorphic Extension from Weakly Pseudoconcave CR Manifolds

Let M be a smooth locally embeddable CR manifold, having some CR dimension m and some CR codimension d. We find an improved local geometric condition on M which guarantees, at a point p on M, that germs of CR distributions are smooth functions, and have extensions to germs of holomorphic functions on a full ambient neighborhood of p. Our condition is a form of weak pseudoconcavity, closely related to essential pseudoconcavity as introduced in [HN1]. Applications are made to CR meromorphic functions and mappings. Explicit examples are given which satisfy our new condition,but which are not pseudoconcave in the strong sense. These results demonstrate that for codimension d > 1, there are additional phenomena which are invisible when d = 1.

Twisting rays imply conformally periodic universes

We point out that algebraically special Einstein fields with twisting rays exhibit some basic properties analogous to the conformal universes considered recently by Roger Penrose.

Leray residues and Abel's theorem inCR codimensionk

In this paper we generalize Lerays calculus of residues in several complex variables, to the situation of an abstract smooth CR manifold M of general type (n,k).

On the Cauchy problem for the \bar\partial operator

We present new results concerning the solvability, or lack of thereof, in the Cauchy problem for the