Differential Geometry and Topology

course: Heat flows for extremal Kaehler metrics

speaker: Santiago R. Simanca (University of New Mexico)

abstract: Let (M,J,w) be a polarized complex manifold of Kaehler type. Let G be the maximal compact subgroup of the automorphism group of (M,J). On the space of Kaehler metrics that are invariant under G and represent the cohomology class w, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the L²-norm of the scalar curvature. We discuss basic properties of the dynamical system in this space of metrics defined by the said flow, and sketch the proof of its local time existence (which, given its pseudo-differential nature, does not follow from previously known results). For complex surfaces of positive first Chern class, this flow exhibits some properties that makes it a good tool to attack the problem of existence of extremal representatives of w. We shall attempt to discuss them in some detail.

timetable:
Fri 1 Oct, 11:15 - 12:15, Sala Conferenze Centro De Giorgi
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