114 research outputs found
A note on the stationary Euler equations of hydrodynamics
This note concerns stationary solutions of the Euler equations for an ideal
fluid on a closed 3-manifold. We prove that if the velocity field of such a
solution has no zeroes and real analytic Bernoulli function, then it can be
rescaled to the Reeb vector field of a stable Hamiltonian structure. In
particular, such a vector field has a periodic orbit unless the 3-manifold is a
torus bundle over the circle. We provide a counterexample showing that the
correspondence breaks down without the real analyticity hypothesis.Comment: 28 pages, no figures, counterexample adde
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
Translated points and Rabinowitz Floer homology
We prove that if a contact manifold admits an exact filling then every local
contactomorphism isotopic to the identity admits a translated point in the
interior of its support, in the sense of Sandon [San11b]. In addition we prove
that if the Rabinowitz Floer homology of the filling is non-zero then every
contactomorphism isotopic to the identity admits a translated point, and if the
Rabinowitz Floer homology of the filling is infinite dimensional then every
contactmorphism isotopic to the identity has either infinitely many translated
points, or a translated point on a closed leaf. Moreover if the contact
manifold has dimension greater than or equal to 3, the latter option
generically doesn't happen. Finally, we prove that a generic contactomorphism
on has infinitely many geometrically distinct iterated
translated points all of which lie in the interior of its support.Comment: 13 pages, v2: numerous corrections, results unchange
Symplectic capacity and short periodic billiard trajectory
We prove that a bounded domain in with smooth boundary has a
periodic billiard trajectory with at most bounce times and of length less
than , where is a positive constant which depends only on
, and is the supremum of radius of balls in . This
result improves the result by C.Viterbo, which asserts that has a
periodic billiard trajectory of length less than C'_n \vol(\Omega)^{1/n}. To
prove this result, we study symplectic capacity of Liouville domains, which is
defined via symplectic homology.Comment: 32 pages, final version with minor modifications. Published online in
Mathematische Zeitschrif
Contact orderability up to conjugation
We study in this paper the remnants of the contact partial order on the
orbits of the adjoint action of contactomorphism groups on their Lie algebras.
Our main interest is a class of non-compact contact manifolds, called convex at
infinity.Comment: 28 pages, 1 figur
Stable Hamiltonian structures in dimension three are supported by open books
We prove that every stable Hamiltonian structure on a closed oriented
three-manifold is stably homotopic to one which is supported (with suitable
signs) by an open book.Comment: 30 pages, 6 figure
Leaf-wise intersections and Rabinowitz Floer homology
In this article we explain how critical points of a particular perturbation
of the Rabinowitz action functional give rise to leaf-wise intersection points
in hypersurfaces of restricted contact type. This is used to derive existence
and multiplicity results for leaf-wise intersection points in hypersurfaces of
restricted contact type in general exact symplectic manifolds. The notion of
leaf-wise intersection points was introduced by Moser.Comment: 18 pages, 1 figure; v3: completely rewritten, improved result
An exact sequence for contact- and symplectic homology
A symplectic manifold with contact type boundary induces
a linearization of the contact homology of with corresponding linearized
contact homology . We establish a Gysin-type exact sequence in which the
symplectic homology of maps to , which in turn maps to
, by a map of degree -2, which then maps to . Furthermore, we
give a description of the degree -2 map in terms of rational holomorphic curves
with constrained asymptotic markers, in the symplectization of .Comment: Final version. Changes for v2: Proof of main theorem supplemented
with detailed discussion of continuation maps. Description of degree -2 map
rewritten with emphasis on asymptotic markers. Sec. 5.2 rewritten with
emphasis on 0-dim. moduli spaces. Transversality discussion reorganized for
clarity (now Remark 9). Various other minor modification
The symplectic Deligne-Mumford stack associated to a stacky polytope
We discuss a symplectic counterpart of the theory of stacky fans. First, we
define a stacky polytope and construct the symplectic Deligne-Mumford stack
associated to the stacky polytope. Then we establish a relation between stacky
polytopes and stacky fans: the stack associated to a stacky polytope is
equivalent to the stack associated to a stacky fan if the stacky fan
corresponds to the stacky polytope.Comment: 20 pages; v2: To appear in Results in Mathematic
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