Model pseudoconvex domains and bumping

The Levi geometry at weakly pseudoconvex boundary points of domains in C^n, n \geq 3, is sufficiently complicated that there are no universal model domains with which to compare a general domain. Good models may be constructed by bumping outward a pseudoconvex, finite-type \Omega \subset C^3 in such a way that: i) pseudoconvexity is preserved, ii) the (locally) larger domain has a simpler defining function, and iii) the lowest possible orders of contact of the bumped domain with \bdy\Omega, at the site of the bumping, are realised. When \Omega \subset C^n, n\geq 3, it is, in general, hard to meet the last two requirements. Such well-controlled bumping is possible when \Omega is h-extendible/semiregular. We examine a family of domains in C^3 that is strictly larger than the family of h-extendible/semiregular domains and construct explicit models for these domains by bumping.Comment: 28 pages; typos corrected; Remarks 2.6 & 2.7 added; clearer proof of Prop. 4.2 given; to appear in IMR

Complex geodesics, their boundary regularity, and a Hardy--Littlewood-type lemma

We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a convex domain $\Omega\Subset \mathbb{C}^n$, $n\geq 2$, is affected by the extent to which $\partial\Omega$ curves or bends at each boundary point. We provide a sufficient condition to this effect (on $\mathcal{C}^1$-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy--Littlewood-type lemma that might be of independent interest.Comment: 10 pages; to appear in Ann. Acad. Sci. Fennicae. Mat

Polynomial approximation, local polynomial convexity, and degenerate CR singularities -- II

We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function (say F) is smooth --- arise from versions of the Weierstrass Approximation Theorem on \bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \bar{D}. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C^2 at an isolated complex tangency.Comment: 11 pages; typos corrected; to appear in Internat. J. Mat

The role of Fourier modes in extension theorems of Hartogs-Chirka type

We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the bidisc. We extend holomorphic functions by applying the Kontinuitaetssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirka's theorem.Comment: 17 page

Rigidity of holomorphic maps between fiber spaces

In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds $X$ and $Y$, a degree-one holomorphic map $f: Y\to X$ is a biholomorphism. Given that the real manifolds underlying $X$ and $Y$ are diffeomorphic, we provide a condition under which $f$ is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products $X=X_1\times X_2$ and $Y=Y_1\times Y_2$ of compact connected complex manifolds. When $X_1$ is a Riemann surface of genus $\geq 2$, we show that any non-constant holomorphic map $F:Y\to X$ is of a special form.Comment: 7 pages; expanded Remark 1.2; provided an explanation for the notation in Section 3; to appear in Internat. J. Mat

The dynamics of holomorphic correspondences of P^1: invariant measures and the normality set

This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if $F$ is a holomorphic correspondence on $\mathbb{P}^1$, then (under certain conditions) $F$ admits a measure $\mu_F$ such that, for any point $z$ drawn from a "large" open subset of $\mathbb{P}^1$, $\mu_F$ is the weak*-limit of the normalised sums of point masses carried by the pre-images of $z$ under the iterates of $F$. Let ${}^\dagger{F}$ denote the transpose of $F$. Under the condition $d_{top}(F) > d_{top}({}^\dagger{F})$, where $d_{top}$ denotes the topological degree, the above phenomemon was established by Dinh and Sibony. We show that the support of this $\mu_F$ is disjoint from the normality set of $F$. There are many interesting correspondences on $\mathbb{P}^1$ for which $d_{top}(F) \leq d_{top}({}^\dagger{F})$. Examples are the correspondences introduced by Bullett and collaborators. When $d_{top}(F) \leq d_{top}({}^\dagger{F})$, equidistribution cannot be expected to the full extent of Brolin's theorem. However, we prove that when $F$ admits a repeller, equidistribution in the above sense holds true.Comment: 24 pages; Section 3 significantly shortened, typos in the proof of Theorem 3.2 removed and Remark 5.3 added; has appeared in Complex Var. Elliptic Equ. as referenced belo

Proper holomorphic maps between bounded symmetric domains revisited

We prove that a proper holomorphic map between two bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively.Comment: 19 pages; typos corrected; missing hypothesis added to the statement of Lemma 4.2; to appear in Pacific J. Mat

Some new observations on interpolation in the spectral unit ball

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;\Omega_n)$-interpolant (D here being the unit disc in the complex plane), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem -- one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of \Omega_n, n\geq 2.Comment: Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/0608177; to appear in Integral Eqns. Operator Theor