An affine sphere equation associated to Einstein toric surfaces
Toshiki Mabuchi
TL;DR
This paper explores a new affine sphere equation linked to Kähler-Einstein metrics on Einstein toric surfaces, extending the understanding of geometric structures in complex differential geometry.
Contribution
It introduces an affine sphere equation associated with Einstein toric surfaces, connecting affine differential geometry with Kähler-Einstein metrics in a novel way.
Findings
Derivation of an affine sphere equation from Kähler-Einstein metrics on Einstein toric surfaces
Discussion of the case for toric surfaces with Kähler-Ricci solitons
Establishment of a natural geometric link between affine spheres and Einstein toric surfaces
Abstract
As seen in the works of Calabi, Cheng-Yau and Loftin, affine sphere equations have a close relationship with Kaehler-Einstein metrics. The main purpose of this note is to show that an equation analogous to those of hyperbolic affine spheres arises naturally from Kaehler-Einstein metrics on Einstein toric surfaces. The case for the remaining toric surfaces with Kaehler-Ricci solitons will also be discussed.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Algebraic Geometry and Number Theory