Rigid fibers of spinning tops

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show that a singular level set of a convex Hamiltonian is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.Comment: 21 pages; some notations in Section 4 change

Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

The first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,\omega_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories.Comment: 24 pages, 6 figure

APPLICATION OF FRAGMENTATION NORMS TO TRANSPORTED POINTS BY HAMILTONIAN ISOTOPIES (Geometry, Algebra and Combinatorics in Transformation group theory)

We prove that a certain C^{0}-robust condition of a Hamiltonian function H induces the existence of a point transported varepsilon out of the original point by the Hamiltonian diffeomorphism varphi_{H}. Related to our observation, we provide a problem on Hamiltonian pseudo-rotations