Energy functionals for Calabi-Yau metrics

TL;DR

This paper introduces energy functionals on Calabi-Yau metrics that enable a new optimization-based numerical approach to approximate Ricci-flat metrics with high accuracy, demonstrated on Fermat quartic and quintics.

Contribution

It develops a set of energy functionals for Calabi-Yau metrics and applies them to improve numerical approximation of Ricci-flat metrics using algebraic metrics.

Findings

Exponential convergence of the method in polynomial degree

More accurate than previous balanced metrics approximations

Applicable to Fermat quartic and quintic Calabi-Yau manifolds

Abstract

We identify a set of "energy" functionals on the space of metrics in a given Kaehler 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 Kaehler 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.