This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface, or, more generally, a symplectic 4-manifold with suitable metric, and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles). We show that the lagrangian homology of a Kähler surface (or of a symplectic 4-manifold) is generated by minimizing lagrangian surfaces that are branched immersions except at finitely many singular points. We precisely describe the structure of these singular points. In particular, these singular points are represented by lagrangian cones with an associated local Maslov index. Only those cones of Maslov index 1 or −1 may be area minimizing. The mean curvature of the minimizers satisfies a first-order system of partial differential equations of "Hodge-type".

We construct isospectral pairs of Riemannian metrics on S⁵ and on B⁶, thus lowering by three the minimal dimension of spheres and balls on which such metrics have been constructed previously (S^n≥8 and B^n≥9). We also construct continuous families of isospectral Riemannian metrics on S⁷ and on B⁸. In each of these examples, the metrics can be chosen equal to the standard metric outside certain subsets of arbitrarily small volume.

The main result of this paper is, "If M is a closed hyperbolic 3-manifold, then the inclusion of the isometry group Isom(M) into the diffeomorphism group Diff(M) is a homotopy equivalence."

A theorem of E. Lerman and S. Tolman, generalizing a result of T. Delzant, states that compact symplectic toric orbifolds are classified by their moment polytopes, together with a positive integer label attached to each of their facets. In this paper we use this result, and the existence of "global" actionangle coordinates, to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold, by means of smooth functions on the corresponding moment polytope. This is equivalent to parametrizing all toric Kähler metrics and generalizes an analogous result for toric manifolds.

A simple explicit description of interesting families of extremal Kähler metrics, arising from recent work of R. Bryant, is given as an application of the approach in this paper. The fact that in dimension four these metrics are self-dual and conformally Einstein is also discussed. This gives rise in particular to a one parameter family of self-dual Einstein metrics connecting the well known Eguchi-Hanson and Taub-NUT metrics.