arXiv 0908.26352009-08hep-thmath.DG

Calabi-Yau度规的能量泛函

Energy functionals for Calabi-Yau metrics

Matthew Headrick, Ali Nassar

备注：: 31 pages; v3: added scan over moduli space including compact conifold and deformed conifold, comparison to Donaldson's refined metrics, references; v2: added computations of two-derivative energy functional; other small changes
发表信息：: Adv. Theor. Math. Phys. 17: 867 (2013)

摘要：

我们确定了一组“能量”泛函的度量空间中的一个给定的Kaehler类的卡-丘流形，这是有界的下面，并最小化唯一的Ricci平坦度量在该类。使用这些泛函，我们重铸的问题，数值求解爱因斯坦方程作为一个优化问题。我们应用这种策略，使用“代数”度量（度量的Kaehler潜在的多项式中的投影坐标），费马四次和一个参数的五次家庭，包括费马和conifold五次。我们表明，这种方法产生近似的Ricci平坦度量是指数准确的多项式的程度（除了在conifold点，收敛是多项式），因此数量级更准确的比平衡的度量，以前研究的近似Ricci平坦度量。该方法相对快速且易于实现。在理论方面，我们还表明，泛函可以用来给一个启发式证明丘的定理。

英文摘要：

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.

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