David E. Barrett

On the smoothness of Levi-foliations

We study the regularity of the induced foliation of a Levi-flat hypersurface in Cn, showing that the foliation is as many times continuously differentiable as the hypersurface itself. The key step in the proof given here is the construction of a certain family of approximate plurisubharmonic defining functions for the hypersurface in question.

Global convexity properties of some families of three-dimensional compact levi-flat hypersurfaces

We consider various examples of compact Levi-flat hypersurfaces in two-dimensional complex manifolds, exploring the interplay between geometric properties of the induced foliation, behavior of the tangential Cauchy-Riemann equations along the hypersurface, and pseudoconvexity properties of small neighborhoods of the hypersurface

Poincaré series and holomorphic averaging

SummaryWe provide an alternate proof of McMullens theorem on contractive properties of the Poincaré series operator in the special case of the universal covering. This case includes in particular Kras Theta Conjecture.

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC∞ boundary such that Ω ≅ ℂ2.

Sums of CR functions from competing CR structures

In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.

Holomorphic motion of circles through affine bundles

The pivotal topic of this paper is the study of Levi-flat real hypersurfaces S with circular fibers in a rank 1 affine bundle A over a Riemann surface X. (To say that S is Levi-flat is to say that S admits a foliation by Riemann surfaces; equivalently, in the language of [SuTh], S may be said to prescribe a holomorphic motion of circles through A.) After setting notation and terminology in §2 we proceed in §3 to examine the Levi-form of a general real hypersurface with circular fibers, emphasizing the connection with curvature considerations. In §4 we focus on the Levi-flat case. In Theorems 5 and 6 we construct moduli spaces for Levi-flat S attached to a fixed underlying line bundle L in the compact and non-compact cases, respectively. In particular, when X is compact we show that the existence of a Levi-flat S implies that 0 ≤ deg L ≤ 2 genus(X) − 2. (The bound is sharp.) Theorem 7 in §7 states that when S is Levi-flat, the Levi-foliation on S extends to a holomorphic foliation of the CP bundle obtained from A by compactifying the fibers. In the general case, the extended foliation in constructed by looking for holomorphic sections of A whose distance from the center is harmonic with respect to the appropriate metric. In §7 we show that this construction produces a foliation even in some cases where S “disappears into the recomplexification of A.” §6 looks at general holomorphic foliations (transverse to fibers) of compactified rank 1 affine bundles; in particular, it is shown that such foliations are classified up to equivalence by a “Schwarzian derivative” and a “curvature function.” An Addendum to Theorem 7 shows how to recognize when such a foliation arises from a Levi-flat hypersurface. The remaining sections contain postponed proofs.

On a class of conformal metrics arising in the work of Seiberg and Witten

Abstract. We examine a class of conformal metrics arising in the “N = 2 supersymmetric Yang-Mills theory” of Seiberg and Witten. We provide several alternate characterizations of this class of metrics and proceed to examine issues of existence and boundary behavior and to parameterize the collection of Seiberg-Witten metrics with isolated non-essential singularities on a fixed compact Riemann surface. In consequence of these results, the Riemann sphere

A generalized Cousin problem for subvarieties of the bidisk

\hat{\mathbb{C}}

Behavior of the Bergman projection on the Diederich-Fornæss worm

does not admit a Seiberg-Witten metric, but for all

Regularity of the Bergman projection on domains with transverse symmetries

\epsilon>0