Energy functionals for Calabi-Yau metrics

TL;DR

The paper introduces energy functionals on the space of Kähler metrics within a fixed CY Kähler class that are bounded below and uniquely minimized by the Ricci-flat metric, recasting the Einstein equation as a variational optimization problem. It couples these functionals with algebraic (degree-k) metrics obtained from an embedding into projective space to produce what they call optimal metrics, which satisfy a Galerkin-type condition approximating the Monge-Ampère equation. Applied to the Fermat quartic and a one-parameter family of quintics, the method achieves exponential convergence in the degree k (except at conifold points where convergence is polynomial) and outperforms balanced metrics by orders of magnitude in accuracy. The approach is computationally efficient, scalable to higher k, and provides a practical framework that could yield high-precision CY metrics and even offer heuristic insight toward Yau's theorem; it also opens avenues for computing geometric quantities and exploring extensions to broader CY and non-CY Kähler settings.

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.