Symplectic capacities of Hermitian symmetric spaces of compact and noncompact type

Abstract

Abstract. Inspired by the work of G. Lu [34] on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer-Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer-Zehnder capacity for Cartan domains and their products.

Some remarks on the Gromov width of homogeneous Hodge manifolds

“…Recently (see [9]) the authors of the present paper have computed the Gromov width of all Hermitian symmetric spaces of compact and noncompact type and their products extending the previous results of G. Lu [10] (see also [8]) for the case of complex Grassmanians.…”

“…In the symmetric case inequalities (4) and (5) are equalities (see [9] for a proof) and we believe that this holds true also in the non symmetric cases. To this respect recall a conjecture due to P. Biran which asserts that π is a lower bound for the Gromov width of any closed integral symplectic manifold.…”

“…In the last twenty years computations and estimates of the Gromov width for various examples have been obtained by several authors (see, e.g. [9] and reference therein).…”

Minimal symplectic atlases of Hermitian symmetric spaces

“…Vol(CP n ) = π n n! and that the Gromov width of any HSSCT (see [14]) is given by c G (M, ω F S ) = π. where Cut p (M ) is the cut locus of (M, ω F S ) with respect to a fixed point p ∈ M (see also [13]). Thus Vol(M, ω F S ) = Vol(Ω, ω 0 ).…”

On the Gromov width of homogeneous Kähler manifolds

“…The class of manifolds in Theorem 1 includes all Hermitian symmetric space of compact type whose Gromov width has been computed in [5]. We do not know if the assumption on the second Betti number can be dropped.…”