---
tag: 6ETEBA8827
kind: concept
name: Kähler–Einstein metric
msc: [32Q20, 53C55, 53C25]
specializes: [41S2EZ0KR3]
sources:
  - url: https://en.wikipedia.org/wiki/K%C3%A4hler%E2%80%93Einstein_metric
same_as:
  wikidata: Q6453393
  wikipedia: 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|41S2EZ0KR3]], 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|DHYSPSYSCB]].

## Explanations

### L1 human

On a complex manifold there are many ways to measure lengths that respect the complex structure.
A Kähler–Einstein metric is one whose curvature, in the Ricci sense, is the same in every direction at every point.
The round sphere, seen as the complex projective line, is an example with positive curvature.
A flat torus is an example with zero curvature, and the hyperbolic disc is one with negative curvature.