## Institute for Mathematical Sciences

## Preprint ims03-02

** Santiago R. Simanca**
* Heat Flows for Extremal K\"ahler Metrics*

Abstract: Let $(M,J,\Omega)$ be a polarized complex manifold of K\"ahler type. Let $G$ be the maximal compact subgroup of the automorphism
group of $(M,J)$. On the space of K\"ahler metrics that are
invariant under $G$ and represent the cohomology class $\Omega$,
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.

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