52 research outputs found
Lagrangian non-squeezing and a geometric inequality
We prove that if the unit codisc bundle of a closed Riemannian manifold
embeds symplectically into a symplectic cylinder of radius one then the length
of the shortest nontrivial closed geodesic is at most half the area of the unit
disc.Comment: minor changes, to appear in Math.
The annulus property of simple holomorphic discs
We show that any simple holomorphic disc admits the annulus property, i.e.,
each interior point is surrounded by an arbitrary small annulus consisting
entirely of injective points. As an application we show that interior
singularities of holomorphic discs can be resolved after slight perturbation of
the almost complex structure. Moreover, for boundary points the analogue
notion, the half-annulus property, is introduced and studied in detail.Comment: 23 pages, no figures, article rewritten, to appear in J. Symplectic
Geo
Linking and closed orbits
We show that the Lagrangian of classical mechanics on a Riemannian manifold
of bounded geometry carries a periodic solution of motion with rescribed
energy, provided the potential satisfies an asymptotic growth condition,
changes sign, and the negative set of the potential is non-trivial in the
relative homology.Comment: We weakened the assumption to be of bounded geometr
- …