Hofer-Zehnder capacity of disc tangent bundles of projective spaces (Q6586627)

For a compact symplectic manifold \((M, \omega)\), possibly with boundary \(\partial M\), the Hofer-Zehnder capacity is defined as follows: \N\[\Nc_{HZ} (M, \omega) = \sup \{\max (H) \mid H : M \longrightarrow \mathbb{R}, \text{ smooth and admissible} \} \N\]\NThe Hofer-Zehnder capacity is a numerical symplectic invariant measuring the symplectic size of a symplectic manifold, but it is fairly hard to compute and unknown in many cases.\N\NThe author computes the Hofer-Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension, considering the Fubini-Study metric in the first case, and the round metric in the second case. More precisely, the following results are proved:\N\NTheorem A. Equip \(\mathbb{C}P^n\) with the Fubini-Study metric and \(\mathbb{R}P^n\) with the round metric. Denote by \(l\) the length of the prime geodesics, then \N\[\Nc_{HZ} (D_1, \mathbb{C}P^n, d\lambda) = l \N\]\Nwhile \N\[\Nc_{HZ} (D_1, \mathbb{R} P^n, d\lambda) = 2l \N\]\NHere \(d\lambda\) is the canonical symplectic structure.\N\NThe author also obtains an explicit computation when one considers the magnetically twisted tangent bundle (Theorem B in the paper).\N\NThe constructions are technical but very well developed throughout the paper.