Inverse mean curvature flow

Builds on Mean curvature Riemannian manifold

Definition

Let $(N, g)$ be a Riemannian manifold of dimension $n + 1$ , let $M$ be a closed manifold of dimension $n$ , and let $T > 0$ . A family of immersions $F \colon M \times [0, T) \to N$ is a solution of inverse mean curvature flow if for every $t \in [0, T)$ :

the hypersurface $M_t = F(M, t)$ has an outward unit normal $\nu(\cdot, t)$ ;
the mean curvature $H(\cdot, t)$ of $M_t$ with respect to $\nu$ , the sum of the principal curvatures, is positive everywhere;
for all $x \in M$ , $\frac{\partial F}{\partial t}(x, t) = \frac{1}{H(x, t)}\, \nu(x, t).$

Remarks

Example: in $N = \R^{n+1}$ take $M_t$ the sphere of radius $r(t) = r_0 e^{t/n}$ about the origin. Its mean curvature is $H = n/r$ , so the speed required by clause 3 is $r/n$ , and indeed $r'(t) = r(t)/n$ . Its area is $e^t$ times the area of $M_0$ . Non-example: a hypersurface with a point where $H \le 0$ fails clause 2, and the flow cannot start from it. A torus of revolution in $\R^3$ with tube radius $r$ and center circle of radius $R < 2r$ has such points on its inner circle. Under the flow the area element is multiplied by $e^t$ in general, since its time derivative is $H \cdot (1/H) = 1$ times itself.

Reading path

To understand this page from the ground up, read these 3 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	`ZZFTPP8JTG`
URI	`m2://ZZFTPP8JTG`
Kind	definition
Writing	W0 stub
MSC	53E10, 53C21
Defines	Inverse mean curvature flow
Uses	Mean curvature, Riemannian manifold
Changed	2 November 2026

Inverse mean curvature flow

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here