In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff-Denjoy theorems for a wide collection of domains.

Abstract. We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain Ω ⊂ C n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with ∂Ω, at the site of the bumping, are explicitly realised. Generally, when Ω ⊂ C n , n ≥ 3, the known methods lead to bumpings with high orders of contact -which are not explicitly known either -at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C 3 . These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.

Abstract. We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to ∂D. This example suggests that continuity at the boundary of the complex geodesics of a convex domain Ω ⋐ C n , n ≥ 2, is affected by the extent to which ∂Ω curves or bends at each boundary point. We provide a sufficient condition to this effect (on 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.

This paper is motivated by Brolin's theorem. The phenomenon we wish to demonstrate is as follows: if F is a holomorphic correspondence on P 1 , then (under certain conditions) F admits a measure µF such that, for any point z drawn from a "large" open subset of P 1 , µF is the weak * -limit of the normalised sums of point masses carried by the pre-images of z under the iterates of F . Let † F denote the transpose of F . Under the condition dtop(F ) > dtop( † F ), where dtop denotes the topological degree, the above phenomenon was established by Dinh and Sibony. We show that the support of this µF is disjoint from the normality set of F . There are many interesting correspondences on P 1 for which dtop(F ) ≤ dtop( † F ). Examples are the correspondences introduced by Bullett and collaborators. When dtop(F ) ≤ dtop( † 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.On the dynamics of multivalued maps between complex manifolds: results of perhaps the broadest scope are established in [11]. We borrow from [11] the following definition.Definition 1.1. Let X 1 and X 2 be two compact complex manifolds of dimension k. We say that Γ is a holomorphic k-chain in X 1 × X 2 if Γ is a formal linear combination of the

Abstract. We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ωn, n ≥ 2. We begin by showing that a known necessary condition for the existence of a O(D; Ωn)-interpolant (D here being the unit disc in C), 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 Ωn, n ≥ 2.