---
tag: 51G4QJZ345
kind: concept
name: Loewner differential equation
msc: [30C35, 30C55]
sources:
  - url: https://en.wikipedia.org/wiki/Loewner_differential_equation
same_as:
  wikidata: Q6666620
  wikipedia: Loewner differential equation
---

## Summary

The Loewner differential equation describes a family of conformal maps that change continuously in time.
Each map sends the complement of a growing set onto a fixed reference domain.
The whole family is driven by a single real or circle-valued function of time, the driving function.
Charles Loewner introduced it to prove the case $n = 3$ of the [[Bieberbach conjecture|ZMRFSYVWQT]].
Louis de Branges used it in the proof of the full conjecture.
With a Brownian motion as driving function it defines the [[Schramm–Loewner evolution|AQ75M2JTB2]].

## Explanations

### L1 human

Picture a crack growing into the upper half-plane from a point on the real axis.
At each time, there is a conformal map that opens the cracked half-plane back into the whole half-plane.
These maps change smoothly in time, and the Loewner equation says how.
The speed of the map at a point depends only on the value of the map there and on where the tip of the crack is sent.
For a straight vertical crack of height $2\sqrt{t}$ at $0$, the map is $z \mapsto \sqrt{z^2 + 4t}$.