## Abstract

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 λ_{1} of surfaces in the symplectic category. In particular we prove that every split symplectic manifold T^{4} × M admits a compatible Riemannian metric whose first eigenvalue is arbitrarily large. On the other hand for Kähler metrics compatible with a given integral symplectic form an upper bound for λ_{1} does exist. The second theme is the study of Hamiltonian symplectic fibrations over S^{2}. 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.

Original language | English |
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Pages (from-to) | 1-17 |

Number of pages | 17 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Volume | 502 |

DOIs | |

State | Published - 1998 |