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Numerical Approximations to Extremal Toric Kähler Metrics with Arbitrary Kähler Class

Part of: Partial differential equations, boundary value problems Global differential geometry

Published online by Cambridge University Press:  10 January 2017

Stuart James Hall  and
Thomas Murphy
Stuart James Hall
Affiliation:
Department of Applied Computing, University of Buckingham, Hunter Street, Buckingham MK18 1EG, UK
Thomas Murphy
Affiliation:
Department of Mathematics, California State University Fullerton, 800 N. State College Boulevard, Fullerton, CA 92831, USA ([email protected])
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Abstract

We develop new algorithms for approximating extremal toric Kähler metrics. We focus on an extremal metric on , which is conformal to an Einstein metric (the Chen–LeBrun–Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric that gives numerical evidence that the Einstein metric is conformally unstable under Ricci flow.

Keywords

extremal Kähler metricEinstein metricRicci flowlinear stability

MSC classification

Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55: Hermitian and Kählerian manifolds 65N35: Spectral, collocation and related methods
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 893 - 910
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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