1,026 research outputs found
Strongly Fillable Contact Manifolds and J-holomorphic Foliations
We prove that every strong symplectic filling of a planar contact manifold
admits a symplectic Lefschetz fibration over the disk, and every strong filling
of the 3-torus similarly admits a Lefschetz fibration over the annulus. It
follows that strongly fillable planar contact structures are also Stein
fillable, and all strong fillings of the 3-torus are equivalent up to
symplectic deformation and blowup. These constructions result from a
compactness theorem for punctured J-holomorphic curves that foliate a convex
symplectic manifold. We use it also to show that the compactly supported
symplectomorphism group on the cotangent bundle of the 2-torus is contractible,
and to define an obstruction to strong fillability that yields a
non-gauge-theoretic proof of Gay's recent nonfillability result for contact
manifolds with positive Giroux torsion.Comment: 44 pages, 2 figures; v.3 has a few significant improvements to the
main results: We now classify all strong fillings and exact fillings of T^3
(without assuming Stein), and also show that a planar contact manifold is
strongly fillable if and only if all its planar open books have monodromy
generated by right-handed Dehn twists. To appear in Duke Math.
Holomorphic Curves in Blown Up Open Books
We use contact fiber sums of open book decompositions to define an infinite
hierarchy of filling obstructions for contact 3-manifolds, called planar
k-torsion for nonnegative integers k, all of which cause the contact invariant
in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to
overtwistedness, while every contact manifold with Giroux torsion also has
planar 1-torsion, and we give examples of contact manifolds that have planar
k-torsion for any but no Giroux torsion, leading to many new examples
of nonfillable contact manifolds. We show also that the complement of the
binding of a supporting open book never has planar torsion. The technical basis
of these results is an existence and uniqueness theorem for J-holomorphic
curves with positive ends approaching the (possibly blown up) binding of an
ensemble of open book decompositions.Comment: This preprint is now superseded by the paper "A Hierarchy of Local
Symplectic Filling Obstructions for Contact 3-Manifolds", arXiv:1009.274
In re Parental Rights as to A.D.L., 133 Nev. Adv. Op. 72 (Oct. 5, 2017)
The Nevada Supreme Court held that (1) requiring a parent to admit guilt to a criminal act in order to maintain his or her parental rights violates that parent’s Fifth Amendment rights; and (2) substantial evidence must demonstrate that terminating parental rights is in the best interest of the children when a parent overcomes the presumptions in NRS 128.109(1)-(2)
On a Prototype of an Optimal Control Problem Governed by Ordinary and Partial Differential Equations
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