Differential geometry Geometric evolution equations Theorem

Gage–Hamilton–Grayson theorem

Builds on Curve-shortening flow Curvature of a curve Convex set Circle

Statement

Let $\gamma_0$ be a smooth embedded closed plane curve that encloses area $A_0$ . Then the curve-shortening flow with initial curve $\gamma_0$ has a unique smooth solution $(\gamma_t)_{0 \le t < T}$ with $T = A_0 / (2\pi)$ , and:

each $\gamma_t$ is embedded;
there is $t_0 < T$ such that $\gamma_t$ bounds a strictly convex region for all $t \in [t_0, T)$ ;
as $t \to T$ , the curves $\gamma_t$ converge to a point $p$ . Rescale $\gamma_t$ by the dilation about $p$ that makes the enclosed area equal to $\pi$ . These rescaled curves converge smoothly to the unit circle centered at $p$ .

Remarks

Gage and Hamilton proved in 1986 that convex curves stay convex and shrink to a round point. Grayson proved in 1987 that every embedded curve becomes convex before it becomes singular, which reduces the general case to the convex one. The value of $T$ comes from the identity $A'(t) = -\int_{\gamma_t} \kappa \, ds = -2\pi$ for embedded curves, since the total curvature of a simple closed curve is $2\pi$ . Embeddedness is needed: an immersed figure-eight curve, or a curve with a small loop, can develop a singularity before it shrinks to a point.

Example: a circle of radius $r_0$ encloses area $\pi r_0^2$ and vanishes at time $r_0^2 / 2 = A_0 / (2\pi)$ , in agreement with the formula.

Going deeper

For a student

The theorem is a model for later results on geometric flows, such as Huisken's theorem on convex hypersurfaces under mean curvature flow and Hamilton's work on Ricci flow. Grayson's step is the hard part, because it must rule out singularities for nonconvex curves. Later proofs by Huisken and by Andrews and Bryan compare distances along the curve with distances in the plane. The flow can also be used to prove geometric inequalities and the existence of closed geodesics on surfaces.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`FVGW1381TZ`
URI	`m2://FVGW1381TZ`
Kind	statement, theorem
Writing	W0 stub
MSC	53E10, 53A04
Status	This copy of the site has no build data. Run `m2 status FVGW1381TZ`.
About	Curve-shortening flow
Uses	Curve-shortening flow, Curvature of a curve, Convex set, Circle
Changed	2 November 2026