Differential geometry Geometric evolution equations Theorem

Huisken's theorem on convex hypersurfaces

Builds on Mean curvature Principal curvature

Statement

Let $n \ge 2$ and let $M_0$ be a compact, connected, smooth hypersurface without boundary embedded in $\R^{n+1}$ . Assume that $M_0$ is uniformly convex, that is, all principal curvatures of $M_0$ are positive at every point. Then:

The mean curvature flow starting from $M_0$ has a unique smooth solution $(M_t)_{0 \le t < T}$ on a maximal time interval with $T < \infty$ .
Each $M_t$ is uniformly convex.
As $t \to T$ , the hypersurfaces $M_t$ shrink to a single point $p \in \R^{n+1}$ .
The rescaled hypersurfaces $(2n(T - t))^{-1/2}(M_t - p)$ converge smoothly to the unit sphere $S^n$ as $t \to T$ .

Remarks

The proof first shows that convexity is preserved by the maximum principle applied to the evolution of the second fundamental form. The key estimate is a pinching estimate: the ratio of the smallest to the largest principal curvature tends to $1$ where the mean curvature blows up. Together with bounds on the derivatives of the curvature, this forces the rescaled surfaces to become round. For a round sphere of radius $r_0$ , the flow gives spheres of radius $r(t) = \sqrt{r_0^2 - 2nt}$ , so $T = r_0^2/(2n)$ and the rescaled surfaces are all the unit sphere. Convexity is needed: a dumbbell, made of two large spheres joined by a thin neck, can pinch off at the neck before it shrinks to a point. The case $n = 1$ of convex curves in the plane is the theorem of Gage and Hamilton, part of the Gage–Hamilton–Grayson theorem.

Going deeper

For a student

Huisken proved this in 1984, for hypersurfaces of dimension at least two. It is the model result for geometric flows that make a shape round. It parallels Hamilton's theorem that the Ricci flow deforms a closed $3$ -manifold with positive Ricci curvature into a round one. The method combines the parabolic maximum principle with curvature pinching estimates. Later work of Huisken and others studies what happens without convexity, where singularities such as neck pinches appear.

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Mean curvature concept
Principal curvature concept
Huisken's theorem on convex hypersurfaces statement

Tag	`14RHHMGF2D`
URI	`m2://14RHHMGF2D`
Kind	statement, theorem
Writing	W0 stub
MSC	53E10, 53C44
Status	This copy of the site has no build data. Run `m2 status 14RHHMGF2D`.
Uses	Mean curvature, Principal curvature
Changed	2 November 2026