---
tag: CR14D9YQTY
kind: concept
name: Ricci flow
msc: ["53E20", 35K55]
sources:
  - url: https://en.wikipedia.org/wiki/Ricci_flow
same_as:
  wikidata: Q1361396
  wikipedia: Ricci flow
---

## Summary

The Ricci flow is the evolution equation $\partial_t g = -2 \operatorname{Ric}(g)$ for a family of [[Riemannian metrics|Q61VFHCH2J]] on a manifold.
It was introduced by Richard Hamilton in 1982.
It is a nonlinear, weakly parabolic system that behaves like a heat equation for the metric, smoothing out irregularities of the curvature.
Grigori Perelman used the Ricci flow with surgery to prove the Poincaré conjecture and Thurston's geometrization conjecture.

## Explanations

### L1 human

Think of a lumpy shape that you let relax, with each region changing its size according to how curved it is.
Under the Ricci flow, regions of positive [[Ricci curvature|4GTNM0WHDH]] shrink and regions of negative Ricci curvature expand.
A round sphere stays round and shrinks to a point in finite time.
The hope, made true in many cases, is that a manifold flows toward a simple standard geometry, which then reveals its shape.