English

The Hofer question on intermediate symplectic capacities

Symplectic Geometry 2015-06-12 v3 Dynamical Systems

Abstract

Roughly twenty five years ago Hofer asked: can the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} be symplectically embedded into B^{2(n-1)}(R) \times \mathbb{R}^2 for some R>0? We show that this is the case if R \geq \sqrt{2^{n-1}+2^{n-2}-2}. We deduce that there are no intermediate capacities, between 1-capacities, first constructed by Gromov in 1985, and n-capacities, answering another question of Hofer. In 2008, Guth reached the same conclusion under the additional hypothesis that the intermediate capacities should satisfy the exhaustion property.

Cite

@article{arxiv.1210.1537,
  title  = {The Hofer question on intermediate symplectic capacities},
  author = {Alvaro Pelayo and San Vu Ngoc},
  journal= {arXiv preprint arXiv:1210.1537},
  year   = {2015}
}

Comments

Main theorems and their proofs unchanged. Results not needed for these proofs removed for clarity (and posted as a separate paper). Title changed to emphasize main results. 21 pages, 7 figures

R2 v1 2026-06-21T22:16:31.494Z