This paper calculates the function c(a) whose value at a is the infimum of the size of a ball that contains a symplectic image of the ellipsoid E(1, a).(Here a ≥ 1 is the ratio of the area of the large axis to that of the smaller axis.) The structure of the graph of c(a) is surprisingly rich. The volume constraint implies that c(a) is always greater than or equal to the square root of a, and it is not hard to see that this is equality for large a. However, for a less than the fourth power τ 4 of the golden ratio, c(a) is piecewise linear, with graph that alternately lies on a line through the origin and is horizontal. We prove this by showing that there are exceptional curves in blow ups of the complex projective plane whose homology classes are given by the continued fraction expansions of ratios of Fibonacci numbers. On the interval [τ 4 , 7] we find c(a) = (a + 1)/3. For a ≥ 7, the function c(a) coincides with the square root except on a finite number of intervals where it is again piecewise linear. The embedding constraints coming from embedded contact homology give rise to another capacity function cECH which may be computed by counting lattice points in appropriate right angled triangles. According to Hutchings and Taubes, the functorial properties of embedded contact homology imply that cECH(a) ≤ c(a) for all a. We show here that cECH(a) ≥ c(a) for all a.

Abstract. We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish the Piunikhin-Salamon-Schwarz isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo and Schwarz. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Hamiltonian diffeomorphism, the existence of infinitely many nontrivial periodic points of a compactly supported Hamiltonian diffeomorphism of a subcritical Stein manifold, old and new cases of the Weinstein conjecture, and, most noteworthy, new existence results for closed orbits of a charge in a magnetic field on almost all small energy levels. We shall also obtain some old and new Lagrangian intersection results. Applications to Hofer's geometry on the group of compactly supported Hamiltonian diffeomorphisms will be given in [19].

A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux's Theorem such a manifold looks locally like an open set in some R 2n ∼ = C n with the standard symplectic formand so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities.Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer [19,20] (although the first capacity was in fact constructed by M. Gromov [40]). Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,97]. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in § 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In § 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In § 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.As illustrated below, many of the quantitative aspects of symplectic geometry can be formulated in terms of symplectic capacities. Of course there are other numerical invariants of symplectic manifolds which could be included in *

wobei das Supremum über all jene a genommen wird für die a E 2n (1, ... , 1, r) symplektisch in (M, w) eingebettet werden kann.

We prove that for every subset A of a tame symplectic manifold (W, ω) meeting a semi-positivity condition, the π 1-sensitive Hofer-Zehnder capacity of A is not greater than four times the stable displacement energy of A, c HZ (A, W) ≤ 4e(A × S 1 , W × T * S 1). This estimate yields almost existence of periodic orbits near stably displaceable energy levels of time-independent Hamiltonian systems. Our main applications are: • The Weinstein conjecture holds true for every stably displaceable hypersurface of contact type in (W, ω). • The flow describing the motion of a charge on a closed Riemannian manifold subject to a non-vanishing magnetic field and a conservative force field has contractible periodic orbits at almost all sufficiently small energies. The proof of the above energy-capacity inequality combines a curve shortening procedure in Hofer geometry with the following detection mechanism for periodic orbits: If the ray {ϕ t F }, t ≥ 0, of Hamiltonian diffeomorphisms generated by a compactly supported time-independent Hamiltonian stops to be a minimal geodesic in its homotopy class, then a non-constant contractible periodic orbit must appear.