Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

Details Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics Calabi flow, Geodesic rays, and uniqueness of constant scalar curvature Kähler metrics

2010-04-12

arxiv.org/abs/1004.2012v2

Mathematics

Differential Geometry

2 Figures. Theorem 5.9 added, providing an analytic proof of a result due to X. Chen and G. Szekelyhidi on the lower bound of

Scientific paper

We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exists globally and converges uniformly to a cscK metric in a polynomial rate. Viewed in a K\"ahler class, the Calabi flow is also shown to be asymptotic to a smooth geodesic ray at infinity. This latter fact is also interesting in the finite dimensional analogue, where we show that the downward gradient flow of the Kempf-Ness function in a semi-stable orbit is asymptotic to the direction of optimal degeneration.

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