O ct 2 00 3 HEAT FLOWS FOR EXTREMAL KÄHLER METRICS

Let (M, J, Ω) be a polarized complex manifold of Kähler type. Let G be the maximal compact subgroup of the automorphism group of (M, J). On the space of Kähler metrics that are invariant under G and represent the cohomology class Ω, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the L 2-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points, or extremal solitons, are extremal metrics. We prove local time existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow-up as time approaches it. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.