2007 Geography of non-Kaehler
symplectic torus actions (with Y. Lin). Submitted July 2007, 11
pp .
Abstract . We study the geography of non-Kaehler manifolds
with symplectic torus actions according to their homological data.
More precisely, we exhibit infinitely many examples of homotopically
inequivalent non-Kaehler symplectic manifolds with large symmetry coming
from a symplectic torus action, and group these examples according
to their first integral homology group and first Betti number.
2007 Reduced phase space and toric variety
coordinatizations of Delzant spaces (with J.J. Duistermaat). Submitted
April 2007, 21 pp. (arXiv:0704.0430).
Abstract . In this note we describe the natural coordinatizations of
a Delzant space defined as a reduced phase space (symplectic geometry view-point)
and give explicit formulas for the coordinate transformations. For each fixed
point of the torus action on the Delzant polytope, we have a maximal coordinatization
of an open cell in the Delzant space which contains the fixed point. This cell
is equal to the domain of definition of one of the natural coordinatizations
of the Delzant space as a toric variety (complex algebraic geometry view-point),
and we give an explicit formula for the toric variety coordinates in terms
of the reduced phase space coordinates. We use considerations in the maximal
coordinate neighborhoods to give simple proofs of some of the basic facts about
the Delzant space, as a reduced phase space, and as a toric variety. These
can be viewed as a first application of the coordinatizations, and serve to
make the presentation more self-contained.
2007 Symplectic actions of two-tori
on four-manifolds. To appear in
Mem. Amer. Math. Soc., 81 pp. (arXiv:Math.SG/0609848).
Abstract . We classify symplectic actions of 2-tori on compact, connected
symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends
results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification
is in terms of a collection of invariants of the topology of the manifold,
of the torus action and of the symplectic form. We construct an explicit model
of such symplectic manifolds with torus actions, defined in terms of these
invariants. We also classify, up to equivariant symplectomorphisms, symplectic
actions of (2n-2)-dimensional tori on 2n-dimensional symplectic manifolds,
when at least one orbit is a (2n-2)-dimensional symplectic submanifold. Then
we show that a 2n-dimensional symplectic manifold M equipped with a free symplectic
action of a (2n-2)-dimensional torus with at least one symplectic orbit is
equivariantly diffeomorphic to M/T x T equipped with the translational action
of T. Thus two such symplectic manifolds are equivariantly diffeomorphic if
and only if their orbit spaces are surfaces of the same genus.
2007 Maximal ball packings of symplectic
toric manifolds (with B. Schmidt). Intern.
Math. Res. Not. Vol. 2007, Article
ID rnm139, 24 pages (arXiv:0704.1033).
Abstract . Let M be a symplectic-toric manifold of dimension at
least four. This paper investigates the so called symplectic ball packing
problem
in the toral equivariant setting. We show that the set of toric symplectic
ball packings of M admits the structure of a convex polytope. Previous work
of the first author shows that up to equivalence, only CP^1 x CP^1 and CP^2
admit density one packings when n=2 and only CP^n admits density one packings
when n>2. In contrast, we show that for a fixed n>=2 and each r in (0, 1),
there are uncountably many inequivalent 2n-dimensional symplectic-toric manifolds
with a maximal toric packing of density r. This result follows from a general
analysis of how the densities of maximal packings change while varying a given
symplectic-toric manifold through a family of symplectic-toric manifolds that
are equivariantly diffeomorphic but not equivariantly symplectomorphic.
2007 Symplectic torus actions with
coisotropic principal orbits (with J.J. Duistermaat). Ann.
Inst. Fourier 57, 7 (2007) 2245-2333 (arXiv:Math.DG/0511676).
Abstract . In this paper we completely classify symplectic actions
of a torus T on a compact connected symplectic manifold M when some, hence
every, principal orbit is a coisotropic submanifold. That is, we construct
an explicit model, defined in terms of certain invariants, of the manifold,
the torus action and the symplectic form. The invariants are invariants of
the topology of the manifold, of the torus action, or of the symplectic form.
In order to deal with symplectic actions which are not Hamiltonian, we develop
new techniques, extending the theory of Atiyah, Guillemin-Sternberg, Delzant,
and Benoist. More specifically, we prove that there is a well-defined notion
of constant vector fields on the orbit space M/T. Using a generalization of
the Tietze-Nakajima theorem to what we call V-parallel spaces, we obtain that
M/T is isomorphic to the Cartesian product of a Delzant polytope with a torus.
We then construct special lifts of the constant vector fields on M/T, in terms
of which the model of the symplectic manifold with the torus action is defined.
2007 Topology of spaces of
equivariant symplectic embeddings. Proc.
Amer. Math. Soc. 35 (2007) 277-288. (arXiv:0704.1033).
Abstract . For an n-dimensional torus T, we compute the homotopy type
of the space of T-equivariant symplectic embeddings from the standard 2n-dimensional
ball of some fixed radius into a 2n-dimensional symplectic toric manifold,
and use this computation to define an integer valued step function on the positive
real numbers which is an invariant of the symplectic toric type of our symplectic
manifold.
2006 A geometric approach to the classification
of the equilibrium shapes of self-gravitating fluids (with D. Peralta). Comm.
Math. Physics. 267 no.1 (2006) 93-115.
Abstract . The classification of the equilibrium shapes that a self-gravitating
fluid can take in a Riemannian manifold is a classical problem in Mathematical
Physics. In this paper it is proved that the equilibrium shapes are isoparametric
submanifolds. Some geometric properties of them are also obtained, e.g. classification
and existence for some Riemannian spaces and relationship with the isoperimetric
problem and the group of isometries of the manifold. Our approach to the
problem is geometrical and allows to study the equilibrium shapes on general
Riemannian spaces.
2006 Toric symplectic ball packing. Top. Appl. (2006) 3633-3644. (arXiv:0704.1034).
Abstract . We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.
2006 Unbounded orbits of dynamical systems (with F.G. Gascon and D. Peralta). Appl. Math. Letters 17 (2004) 253-259.
Abstract. We show that when a divergence-free vector field X, without
zeros, is defined on a two-dimensional, noncompact smooth manifold, which is
not an
infinite cylinder, then X must possess an unbounded orbit. The paper focuses
on the description of situations of electromagnetic fields to which the previous
assumptions apply, and hence so does the conclusion.