  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 $lambda_1$ of surfaces in symplectic category. In particular we
prove that every split symplectic manifold $T^4 times M$ 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 $lambda_1$ 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.
