Kähler–Einstein metric

Summary

A Kähler–Einstein metric is a Kähler metric whose Ricci curvature is a constant multiple of the metric. It is both a Kähler metric and an Einstein metric, so it is a canonical choice of metric on a complex manifold. On a compact manifold its existence forces the first Chern class to be negative, zero or positive, and the three cases behave very differently. The negative case was settled by Aubin and by Yau, and the zero case by Yau; the positive case leads to the Yau–Tian–Donaldson conjecture.