Mean curvature flow

Builds on Mean curvature Normal vector Smooth manifold

Definition

Let $M$ be a smooth $n$ -dimensional manifold and let $T > 0$ . A mean curvature flow is a smooth map $F \colon M \times [0, T) \to \R^{n+1}$ such that:

for each $t \in [0, T)$ , the map $F_t = F(\cdot, t)$ is an immersion of $M$ as a hypersurface;
$\partial_t F(p, t) = \vec{H}(p, t)$ for all $p \in M$ and $t \in [0, T)$ .

Here $\vec{H}(p, t) = -H \nu$ is the mean curvature vector of $F_t$ at $p$ . The vector $\nu$ is a unit normal vector, and $H$ is the mean curvature, the sum of the principal curvatures with respect to $\nu$ .

Remarks

Example: let $F(p, t) = R(t)\, p$ for $p$ in the unit sphere $S^n \subset \R^{n+1}$ , with $R(t) = \sqrt{R_0^2 - 2nt}$ . The sphere of radius $R$ has mean curvature vector of length $n/R$ pointing to the centre, so the flow condition is $R'(t) = -n/R(t)$ . This holds since $\frac{d}{dt} R^2 = -2n$ , and the flow exists until $T = R_0^2/(2n)$ . Non-example: $R(t) = R_0 + t$ moves the sphere outward, against the mean curvature vector, so it fails clause 2. A plane or any minimal hypersurface, with $\vec H = 0$ , is a stationary flow. Sign conventions for $H$ and $\nu$ vary, but $\vec H$ always points toward the centre of a round sphere.

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Mean curvature concept
Normal vector concept
Smooth manifold concept
Mean curvature flow definition

Tag	`2GFFGE6WK1`
URI	`m2://2GFFGE6WK1`
Kind	definition
Writing	W0 stub
MSC	53E10, 35K55
Defines	Mean curvature flow
Uses	Mean curvature, Normal vector, Smooth manifold

Mean curvature flow

Definition

Remarks

Reading path

Further reading

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