Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibered spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.Comment: 37 pages, 2 figures; Improved exposition incorporating referee's comments mainly in Sections 5 and 9. (To appear in Geom. Topol.

Effective crustal permeability controls fault evolution: An integrated structural, mineralogical and isotopic study in granitic gneiss, Monte Rosa, Northern Italy

Two dextral faults within granitic gneiss in the Monte Rosa nappe, northern Italy reveal key differences in their evolution controlled by evolving permeability and water/rock reactions. The comparison reveals that identical host rock lithologies develop radically different mineralogies within the fault zones, resulting in fundamentally different deformation histories. Oxygen and hydrogen isotope analyses coupled to microstructural characterisation show that infiltration of meteoric water occurred into both fault zones. The smaller Virgin Fault shows evidence of periodic closed system behaviour, which promoted the growth of hydrothermal K-feldspar, whilst the more open system behaviour of the adjacent Ciao Ciao Fault generated a weaker muscovite-rich fault core, which promoted a step change in fault evolution. Effective crustal permeability is a vital control on fault evolution and, coupled to the temperature (i.e. depth) at which key mineral transformations occur, is probably a more significant factor than host rock strength in controlling fault development. The study suggests that whether a fault in granitic basement grows into a large structure may be largely controlled by the initial hydrological properties of the host rocks. Small faults exposed at the surface may therefore be evolutionary “dead-ends” that typically do not represent the early stages in the development of larger faults

Maps on 3-manifolds given by surgery

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of stable maps from M to the plane in terms of properties of L.Comment: 22 pages, 13 figures. Accepted for publication by Pacific J. Mat

Foliated Structure of The Kuranishi Space and Isomorphisms of Deformation Families of Compact Complex Manifolds

Consider the following uniformization problem. Take two holomorphic (parametrized by some analytic set defined on a neighborhood of $0$ in $\Bbb C^p$, for some $p>0$) or differentiable (parametrized by an open neighborhood of $0$ in $\Bbb R^p$, for some $p>0$) deformation families of compact complex manifolds. Assume they are pointwise isomorphic, that is for each point $t$ of the parameter space, the fiber over $t$ of the first family is biholomorphic to the fiber over $t$ of the second family. Then, under which conditions are the two families locally isomorphic at 0? In this article, we give a sufficient condition in the case of holomorphic families. We show then that, surprisingly, this condition is not sufficient in the case of differentiable families. We also describe different types of counterexamples and give some elements of classification of the counterexamples. These results rely on a geometric study of the Kuranishi space of a compact complex manifold

Geodesic systems of tunnels in hyperbolic 3-manifolds

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a system of at least two tunnels, such that all but one of the tunnels come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number n manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a (1;n)-compression body with a system of core tunnels such that all but one of the core tunnels self-intersect.Comment: 19 pages, 4 figures. V2 contains minor updates to references and exposition. To appear in Algebr. Geom. Topo