22 research outputs found
Local properties of J-complex curves in Lipschitz-continuous structures
We prove the existence of primitive curves and positivity of intersections of
-complex curves for Lipschitz-continuous almost complex structures. These
results are deduced from the Comparison Theorem for -holomorphic maps in
Lipschitz structures, previously known for of class . We also
give the optimal regularity of curves in Lipschitz structures. It occurs to be
, i.e. the first derivatives of a -complex curve for Lipschitz
are Log-Lipschitz-continuous. A simple example that nothing better can be
achieved is given. Further we prove the Genus Formula for -complex curves
and determine their principal Puisieux exponents (all this for
Lipschitz-continuous -s).Comment: Minor corrections and improvements. One example added. To appear in
Math. Zeitschrift
Regular homotopy of Hurwitz curves
We prove that any two irreducible cuspidal Hurwitz curves and (or
more generally, curves with A-type singularities) in the Hirzebruch surface
with coinciding homology classes and sets of singularities are regular
homotopic; and symplectically regular homotopic if and are
symplectic with respect to a compatible symplectic form.Comment: 26 page
Homology class of a Lagrangian Klein bottle
It is shown that an embedded Lagrangian Klein bottle represents a non-trivial
mod 2 homology class in a compact symplectic four-manifold with
. (In versions 1 and 2, the last assumption was missing.
A counterexample to this general claim and the first proof of the corrected
result have been found by Vsevolod Shevchishin.) As a corollary one obtains
that the Klein bottle does not admit a Lagrangian embedding into the standard
symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 -
minor edits, added references; AMSLaTeX, 6 page
Two-dimensional superintegrable metrics with one linear and one cubic integral
We describe all local Riemannian metrics on surfaces whose geodesic flows are
superintegrable with one integral linear in momenta and one integral cubic in
momenta.
We also show that some of these metrics can be extended to the 2-sphere. This
gives us new examples of Hamiltonian systems on the sphere with integrals of
degree three in momenta, and the first examples of superintegrable metrics of
nonconstant curvature on a closed surfaceComment: 35 page
Lagrangian Klein bottles in R^{2n}
It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding
into R^{2n} if and only if n is odd.Comment: V.2 - explicit formula for the Luttinger-type surgery; V.3 - section
3 corrected, section 6 expanded; 6 page
Intersections of quadrics, moment-angle manifolds, and Hamiltonian-minimal Lagrangian embeddings
We study the topology of Hamiltonian-minimal Lagrangian submanifolds N in C^m
constructed from intersections of real quadrics in a work of the first author.
This construction is linked via an embedding criterion to the well-known
Delzant construction of Hamiltonian toric manifolds. We establish the following
topological properties of N: every N embeds as a submanifold in the
corresponding moment-angle manifold Z, and every N is the total space of two
different fibrations, one over the torus T^{m-n} with fibre a real moment-angle
manifold R, and another over a quotient of R by a finite group with fibre a
torus. These properties are used to produce new examples of Hamiltonian-minimal
Lagrangian submanifolds with quite complicated topology.Comment: 14 pages, published version (minor changes