304 research outputs found
Dual elliptic structures on CP2
We consider an almost complex structure J on CP2, or more generally an
elliptic structure E which is tamed by the standard symplectic structure. An
E-curve is a surface tangent to E (this generalizes the notion of
J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that
the space of E-lines is again a CP2 with a tame elliptic structure E^*, and
that each E-curve has an associated dual E^*-curve. This implies that the
E-curves, and in particular the J-curves, satisfy the Pl\"ucker formulas, which
restricts their possible sets of singularities.Comment: 18 pages The only difference with the first version is the mention of
the thesis of Benjamin MacKay ("Duality and integrable systems of
pseudoholomorphic curves", Duke University, 1999), which I did not know at
the time, and which contains a large part of the results of my pape
Bounds on primitives of differential forms and cofilling inequalities
We prove that on a Riemannian manifold, a smooth differential form has a
primitive with a given (functional) upper bound provided the necessary weighted
isoperimetric inequalities implied by Stokes are satisfied. We apply this to
prove a comparison predicted by Gromov between the cofilling function and the
filling area.Comment: The new features of the main result are its sharpness and the fact
that the manifold is not assumed have bounded geometry, nor even to be
complete. This paper corresponds to a part of a talk given in January 2004 in
Haifa, at a workshop in memory of Robert Brooks. The other part, which is the
"translation" in the framework of geometric group theory, will soon be
deposited on arxi
Homological stability properties of spaces of rational J-holomorphic curves in P^2
In a well known work [Se], Graeme Segal proved that the space of holomorphic
maps from a Riemann surface to a complex projective space is homology
equivalent to the corresponding continuous mapping space through a range of
dimensions increasing with degree. In this paper, we address if a similar
result holds when other (not necessarily integrable) almost complex structures
are put on projective space. We take almost complex structures that are
compatible with the underlying symplectic structure. We obtain the following
result: the inclusion of the space of based degree k J-holomorphic maps from
P^1 to P^2 into the double loop space of P^2 is a homology surjection for
dimensions j<3k-2. The proof involves constructing a gluing map analytically in
a way similar to McDuff and Salamon in [MS] and Sikorav in [S] and then
comparing it to a combinatorial gluing map studied by Cohen, Cohen, Mann, and
Milgram in [CCMM].Comment: 28 page
On iterated translated points for contactomorphisms of R^{2n+1} and R^{2n} x S^1
A point q in a contact manifold is called a translated point for a
contactomorphism \phi, with respect to some fixed contact form, if \phi (q) and
q belong to the same Reeb orbit and the contact form is preserved at q. The
problem of existence of translated points is related to the chord conjecture
and to the problem of leafwise coisotropic intersections. In the case of a
compactly supported contactomorphism of R^{2n+1} or R^{2n} x S^1 contact
isotopic to the identity, existence of translated points follows immediately
from Chekanov's theorem on critical points of quasi-functions and Bhupal's
graph construction. In this article we prove that if \phi is positive then
there are infinitely many non-trivial geometrically distinct iterated
translated points, i.e. translated points of some iteration \phi^k. This result
can be seen as a (partial) contact analogue of the result of Viterbo on
existence of infinitely many iterated fixed points for compactly supported
Hamiltonian symplectomorphisms of R^{2n}, and is obtained with generating
functions techniques in the setting of arXiv:0901.3112.Comment: 10 pages, revised version. I removed the discussion on linear growth
of iterated translated points, because it contained a mistake. To appear in
the International Journal of Mathematic
- …