---
tag: 9FBJED00CP
kind: concept
name: Calabi flow
msc: ["53E40", "53C55"]
sources:
  - url: https://en.wikipedia.org/wiki/Calabi_flow
same_as:
  wikidata: Q5018295
  wikipedia: Calabi flow
---

## Summary

The Calabi flow is an evolution of Kähler metrics on a compact complex manifold that stays inside one Kähler class.
It moves the Kähler potential with speed equal to the difference between the [[scalar curvature|ZHX2VYQDZ3]] and its average.
Eugenio Calabi introduced it in his work on extremal Kähler metrics.
Along the flow the Calabi energy, the squared $L^2$ norm of that difference, does not increase.
Its fixed points are the Kähler metrics of constant scalar curvature, so it is a tool for finding such metrics.

## Explanations

### L1 human

A Kähler metric is a way to measure lengths on a complex manifold that fits well with the complex structure.
The Calabi flow changes such a metric step by step, pushing it where the scalar curvature is above average and pulling it where it is below average.
The aim is to reach a metric whose scalar curvature is the same at every point.
The equation is of fourth order in the space variables, so it is harder to study than second-order flows such as the Ricci flow.
A flat metric on a complex torus does not move at all, since its scalar curvature is $0$ everywhere.