Symplectic aspects of the first eigenvalue

There are two themes in the present paper. The first one is spelled out in the title, and is inspired by an attempt to find an analogue of Hersch-Yang-Yau estimate for $lambd{a}_1$ of surfaces in symplectic category. In particular we prove that every split symplectic manifold $T^{4}timesM$ admits a compatible Riemannian metric whose first eigenvalue is arbitrary large. On the other hand for Kahler metrics compatible with a given integral symplectic form an upper bound for $lambd{a}_1$ does exist. The second theme is the study of Hamiltonian symplectic fibrations over the 2-sphere. We construct a numerical invariant called the size of a fibration which arises as the solution of certain variational problems closely related to Hofer’s geometry, K-area and coupling. In some examples it can be computed with the use of Gromov-Witten invariants. The link between these two themes is given by an observation that the first eigenvalue of a Riemannian metric compatible with a symplectic fibration admits a universal upper bound in terms of the size.