Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs
Jaime BustilloÉcole Normale Supérieure, Paris, France
In the first part of the article we study Hamiltonian diffeomorphisms of which are generated by sub-quadratic Hamiltonians and prove a middle dimensional rigidity result for the image of coisotropic cylinders. The tools that we use are Viterbo’s symplectic capacities and a series of inequalities coming from their relation with symplectic reduction. In the second part we consider the nonlinear string equation and treat it as an infinite-dimensional Hamiltonian system. In this context we are able to apply Kuksin’s approximation by finite dimensional Hamiltonian flows and prove a PDE version of the rigidity result for coisotropic cylinders. As a particular example, this result can be applied to the sine-Gordon equation.
Cite this article
Jaime Bustillo, Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs. Comment. Math. Helv. 94 (2019), no. 4, pp. 803–832DOI 10.4171/CMH/474