The Scalar-flat Kaehler Metric and Painleve III
Shoji Okumura
TL;DR
This paper investigates the anti-self-dual equation for SU(2)-invariant metrics, deriving a ninth-order system that simplifies to a sixth-order system for scalar-flat-Kaehler metrics, advancing understanding of special geometric structures.
Contribution
It introduces a ninth-order system for anti-self-dual SU(2)-invariant metrics and reduces it to a sixth-order system for scalar-flat-Kaehler metrics, providing new analytical tools.
Findings
Derived a ninth-order system for anti-self-dual SU(2)-invariant metrics
Reduced the system to sixth-order for scalar-flat-Kaehler metrics
Established an equivalent formulation for these geometric structures
Abstract
We study the anti-self-dual equation for non-diagonal SU(2)-invariant metrics and give an equivalent ninth-order system. This system reduce to a sixth-order system if the metric is in the conformal class of scalar-flat-Kaehler metric.
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Taxonomy
TopicsGeometry and complex manifolds · Advanced Differential Geometry Research · Geometric Analysis and Curvature Flows