The mean curvature vector is a normal vector field obtained by tracing the second fundamental
form of an immersed submanifold. In the trace
convention used for mean curvature flow, an
immersion 
into Euclidean space has
 |
where 
is the divergence-of-gradient
convention on the submanifold. For a surface,
if scalar mean curvature is the average of its
two principal curvatures, then 
with a consistently chosen unit normal and sign convention.
For a round 
-hypersphere of radius 
, this vector
points inward and has length 
. The flow with normal velocity 
shrinks hyperspheres
and has self-shrinkers as dilation solutions.
