Wiman–Valiron Discs and the Dimension of Julia Sets

We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

On n-th roots of meromorphic maps

Let S be a connected Riemann surface and let φ: S → Ĉ bebranched covering map of nite type. If n ≥ 2,then we describe a simple geometrical necessary and sucient condition for the existence of some n-th root, that is, a meromorphic map ψ: S → Ĉ such that φ = ψn.

Fatou components and singularities of meromorphic functions

We prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

Lagrangian Constant Cycle Subvarieties in Lagrangian Fibrations

We show that the image of a dominant meromorphic map from an irreducible compact Calabi–Yau manifold X whose general fiber is of dimension strictly between 0 and $\dim X$ is rationally connected. Using this result, we construct for any hyper-Kähler manifold X admitting a Lagrangian fibration a Lagrangian constant cycle subvariety ΣH in X which depends on a divisor class H whose restriction to some smooth Lagrangian fiber is ample. If $\dim X = 4$, we also show that up to a scalar multiple, the class of a zero-cycle supported on ΣH in CH0(X) depend neither on H nor on the Lagrangian fibration (provided b2(X) ≥ 8).

CR-MEROMORPHIC EXTENSION AND THE NONEMBEDDABILITY OF THE ANDREOTTI–ROSSI CR STRUCTURE IN THE PROJECTIVE SPACE

Let [Formula: see text] be a polynomially convex compact set and let M be a (2p-1) dimensional (p ≥ 2) maximally complex bounded scarred C1 submanifold of [Formula: see text], irreducible in the current sense. According to Harvey–Lawson [14] and Chirka [4], there exists a bounded irreducible analytic set [Formula: see text] such that [M]=±d[T]. In this paper, we prove that every CR-meromorphic map carrying M into a projective manifold V extends to a meromorphic map F:T → V. We extend the notion of CR-meromorphic maps to CR submanifolds of [Formula: see text] and give another proof of our extension theorem which extends to the greater codimensional case. We also apply our extension result to prove a Lewy type extension theorem for CR-meromorphic maps, a Hartogs type theorem in [Formula: see text] and the non embedding of the Andreotti–Rossi CR structure in [Formula: see text].