Advances in Theoretical and Mathematical Physics

Energy functionals for Calabi-Yau metrics

Matthew Headrick and Ali Nassar

Full-text: Open access

Abstract

We identify a set of “energy” functionals on the space of metrics in a given Kähler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the “algebraic” metrics (metrics for which the Kähler potential is given in terms of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We show that this method yields approximations to the Ricci-flat metric that are exponentially accurate in the degree of the polynomial (except at the conifold point, where the convergence is polynomial), and therefore orders of magnitude more accurate than the balanced metrics, previously studied as approximations to the Ricci-flat metric. The method is relatively fast and easy to implement. On the theoretical side, we also show that the functionals can be used to give a heuristic proof of Yau’s theorem.

Article information

Source
Adv. Theor. Math. Phys., Volume 17, Number 5 (2013), 867-902.

Dates
First available in Project Euclid: 21 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1408626508

Mathematical Reviews number (MathSciNet)
MR3251826

Zentralblatt MATH identifier
1304.14050

Citation

Headrick, Matthew; Nassar, Ali. Energy functionals for Calabi-Yau metrics. Adv. Theor. Math. Phys. 17 (2013), no. 5, 867--902. https://projecteuclid.org/euclid.atmp/1408626508


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