The mean curvature flow evolves an immersed submanifold in the direction of its mean curvature vector.
For an immersion 
into Euclidean space, a
parametrization with purely normal velocity satisfies
 |
Here 
uses the trace of the second fundamental form;
for a surface it is twice the scalar mean
curvature times the chosen normal, with the corresponding sign convention. A
round 
-hypersphere in 
shrinks with radius satisfying
.
A self-shrinker gives a solution obtained by dilation.
