The Weinstein Conjecture for Planar Contact Structures in Dimension Three

In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.Comment: 19 pages, corrected some typo

A note on Reeb dynamics on the tight 3-sphere

We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.Comment: 20 pages, no figure

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

Families of Riemann Surfaces, Uniformization and Arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2-dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read off from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk