Twistor construction of Einstein metrics.

*(English)*Zbl 0615.53057
Global Riemannian geometry, Proc. Symp., Durham/Engl. 1982, 115-125 (1984).

[For the entire collection see Zbl 0614.00017.]

One of the most intriguing and important consequences of Yau’s proof of the Calabi conjecture [S. T. Yau, Proc. Natl. Acad. Sci. USA 74, 1798-1799 (1977; Zbl 0355.32028)] is the following theorem:

Theorem 1.1 (Yau). Let X be a Kählerian K3 surface, and \(\alpha \in H^{1,1}(X,{\mathbb{R}})\) a positive cohomology class. Then there exists a unique Kähler metric on X such that the Ricci tensor vanishes, and \(\alpha\) is the class defined by the Kähler form.

The proof of this theorem consists of an existence theorem for certain non-linear partial differential equations, the complex Monge-Ampère equations, and the end result is of importance not only to differential geometers, but also to those who work in algebraic geometry and in general relativity [D. Page, Phys. Lett. B 80, 55-57 (1978)]. The question Yau’s theorem provokes is clear:

Problem. Find the K3 metric explicitly.

The aim of this lecture is not to solve this problem (would that it were!) but first to describe how Penrose’s twistor theory gies a method of attacking it, and secondly to show how that method may be used to provide explicit approximations to the K3 metric. The existence of one of these approximations was deduced heuristically by D. Page [Phys. Lett. B 100, 313-315 (1981)] as a degeneration of the K3 metric.

One of the most intriguing and important consequences of Yau’s proof of the Calabi conjecture [S. T. Yau, Proc. Natl. Acad. Sci. USA 74, 1798-1799 (1977; Zbl 0355.32028)] is the following theorem:

Theorem 1.1 (Yau). Let X be a Kählerian K3 surface, and \(\alpha \in H^{1,1}(X,{\mathbb{R}})\) a positive cohomology class. Then there exists a unique Kähler metric on X such that the Ricci tensor vanishes, and \(\alpha\) is the class defined by the Kähler form.

The proof of this theorem consists of an existence theorem for certain non-linear partial differential equations, the complex Monge-Ampère equations, and the end result is of importance not only to differential geometers, but also to those who work in algebraic geometry and in general relativity [D. Page, Phys. Lett. B 80, 55-57 (1978)]. The question Yau’s theorem provokes is clear:

Problem. Find the K3 metric explicitly.

The aim of this lecture is not to solve this problem (would that it were!) but first to describe how Penrose’s twistor theory gies a method of attacking it, and secondly to show how that method may be used to provide explicit approximations to the K3 metric. The existence of one of these approximations was deduced heuristically by D. Page [Phys. Lett. B 100, 313-315 (1981)] as a degeneration of the K3 metric.