---
tag: Z8192111H3
kind: definition
name: Curve-shortening flow
msc: ["53E10", 53A04]
defines: DVGFRRYK10
uses: [7JQS6VKFEZ]
sources:
  - url: https://en.wikipedia.org/wiki/Curve-shortening_flow
---

## Definition

Let $S^1 = \R / \Z$ and let $T > 0$.
Let $\gamma \colon S^1 \times [0, T) \to \R^2$ be continuous and smooth on $S^1 \times (0, T)$.
Assume that for each $t \in (0, T)$ the curve $\gamma_t = \gamma(\cdot, t)$ is an immersed plane curve.
Let $\kappa(u, t)$ be the signed [[curvature|7JQS6VKFEZ]] of $\gamma_t$ at $u$.
Let $N(u, t)$ be the unit normal obtained by turning the unit tangent by $+90^\circ$.
The family $\gamma$ is a solution of the *curve-shortening flow* with initial curve $\gamma_0$ if
$$\frac{\partial \gamma}{\partial t}(u, t) = \kappa(u, t)\, N(u, t) \qquad \text{for all } u \in S^1, \ t \in (0, T).$$

## Remarks

Equivalently, $\partial_t \gamma = \partial_s^2 \gamma$, where $s$ is the arc length parameter of $\gamma_t$; this is a nonlinear partial differential equation of heat type.
The normal velocity is what matters: adding a tangential velocity only changes the parametrization of the curves.
The length $L(t)$ of $\gamma_t$ satisfies $L'(t) = -\int_{\gamma_t} \kappa^2 \, ds$, and the enclosed area $A(t)$ of an embedded counterclockwise curve satisfies $A'(t) = -2\pi$.

Example: the circles $\gamma(u, t) = \sqrt{r_0^2 - 2t}\,(\cos 2\pi u, \sin 2\pi u)$ for $0 \le t < r_0^2 / 2$ form a solution.
For a counterclockwise circle of radius $r$, $\kappa = 1/r$ and $N$ points to the center, and indeed $\partial_t r = -1/r$.

Non-example: the shrinking circles $\gamma(u, t) = (r_0 - t)(\cos 2\pi u, \sin 2\pi u)$ move with normal speed $1$ instead of $1/r$, so they do not solve the equation.