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.

International Water Rights on the White Nile of the New State of South Sudan

The birth of South Sudan falls directly in the demarcation zone of the rivalry between downstream and upstream riparian states on the waters of the Nile River. The downstream states—Egypt and Sudan—stress their “natural and historic” rights to the entire flow of the Nile based on the 1959 Nile Agreement and older colonial treaties, while the upstream African states refuse to be bound by colonial treaties and claim their equitable share of the Nile River by promoting South Sudan’s accession to the Cooperative Framework Agreement (CFA). The Nile River Basin lacks an international binding water agreement that includes and satisfies all the riparian states. This Article analyzes the status quo of South Sudan’s water rights to the Nile River by addressing the following questions: Is the new state bound by any rights and obligations established by the 1959 Nile Agreement? Is it advantageous for South Sudan to accede to the CFA, which provides for modern principles of international water law? The Article applies the customary international law of state succession to South Sudan’s secession from Sudan to determine if the 1959 Nile Agreement is binding between the two states. It concludes that South Sudan succeeded Sudan with regard to territorial rights and obligations established by the 1959 Nile Agreement, as customary international law recognizes that legal obligations of a territorial nature remain unaffected by state succession. South Sudan should enter into negotiations on a binding water agreement to allocate the 18.5 billion cubic meters of water granted to it under the 1959 Nile Agreement. The Article concludes that South Sudan should accede to the CFA within its allotted portion of the Nile waters under the 1959 Nile Agreement

Algebraic Torsion in Contact Manifolds

We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic torsion of order zero if and only if it is algebraically overtwisted (i.e. has trivial contact homology), and any contact 3-manifold with positive Giroux torsion has algebraic torsion of order one (though the converse is not true). We also construct examples for each nonnegative k of contact 3-manifolds that have algebraic torsion of order k but not k - 1, and derive consequences for contact surgeries on such manifolds. The appendix by Michael Hutchings gives an alternative proof of our cobordism obstructions in dimension three using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a final update to agree with the published paper, and also corrects a minor error that appeared in the published version of the appendi

Contact Hypersurfaces in Uniruled Symplectic Manifolds Always Separate

We observe that nonzero Gromov-Witten invariants with marked point constraints in a closed symplectic manifold imply restrictions on the homology classes that can be represented by contact hypersurfaces. As a special case, contact hypersurfaces must always separate if the symplectic manifold is uniruled. This removes a superfluous assumption in a result of G. Lu, thus implying that all contact manifolds that embed as contact type hypersurfaces into uniruled symplectic manifolds satisfy the Weinstein conjecture. We prove the main result using the Cieliebak-Mohnke approach to defining Gromov-Witten invariants via Donaldson hypersurfaces, thus no semipositivity or virtual moduli cycles are required.Comment: 24 pages, 1 figure; v.3 is a substantial expansion in which the semipositivity condition has been removed by implementing Cieliebak-Mohnke transversality; it also includes a new appendix to explain why the forgetful map in the Cieliebak-Mohnke context is a pseudocycle; v.4 has one short remark added; to appear in J. London Math. So