arXiv subjects
Publications and source records attributed to Morimichi Kawasaki.
Hofer's norm (metric) is an important and interesting topic in symplectic geometry. In the present paper, we define fragmented Hofer's norms which are Hofer's norms controlled by fragmentation norms and give some observations on fragmented Hofer's norms.
Stefan M$\ddot{\mathrm{u}}$ller posed the problem "Do Hofer's metrics on the group of Hamiltonian diffeomorphism and the one of Hamiltonian homeomorphisms (Hameomorphisms) correspond?". Let $(M,ω)$ be a compact exact symplectic manifold. We prove that the group of Hamiltonian homeomorphisms is not a simple group if the positive answer of M$\ddot{\mathrm{u}}$ller's question holds . To prove this results, we introduce fragmented Hofer's norms on a normal subgroup of the group of Hamiltonian homeomorphisms.
Entov and Polterovich defined heaviness for closed subsets of a symplectic manifold by using the Hamiltonian Floer theory on contractible trajectories. Heavy subsets are known to be non-displaceable. In the present paper, we define a relative symplectic capacity $C(M,X,R;e)$ for a symplectic manifold $(M,ω)$ and its subset $X$ which measures the existence of non-contractible trajectories of Hamiltonian isotopies on the product with annulus. We prove that $C(M,X,R;e)$ is finite if $(M,ω)$ is monotone and $X$ is a heavy subset. We also prove that $C(M,X,R;e)$ is infinite if $X$ is a displaceable compact subset.
We show that the union of a meridian and a longitude of the symplectic 2-torus is superheavy in the sense of Entov-Polterovich. By a result of Entov-Polterovich, this implies that the product of this union and the Clifford torus of $\mathbb{C}P^n$ with the Fubini-Study symplectic form cannot be displaced by symplectomorphisms.
Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which are geralizations of quasimorphisms. In the present paper, we prove a Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms on stable groups. %We also give an application of its generalization to symplectic geometry related to Entov-Polterovich's theory on heavy subsets. We also pose a generalization of our main theorem and prove that "stably non-displaceable subsets of symplectic manifolds are heavy" in a rough sense if that generalization holds.