Energy functionals for Calabi-Yau metrics

We identify a set of “energy” functionals on the space of metrics in a given Kähler class on a Calabi-Yau manifold, that 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 test this strategy using the “algebraic” metrics (metrics for which the Kähler potential is given in terms of a polynomial in the projective coordinates), showing that they yield approximations to the Ricci-flat metric that are exponentially accurate in the degree of the polynomial, and 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. ar X iv :0 90 8. 26 35 v1 [ he pth ] 1 9 A ug 2 00 9