The Willmore energy of a smooth immersion of a closed surface 
into 3-dimensional Euclidean space is
 |
where 
is the mean curvature, defined as the average of
the two principal curvatures, and 
is the area element. Some
authors instead use 
,
where 
is the Gaussian curvature. For closed
surfaces, the two conventions differ only by a topological constant by the Gauss-Bonnet formula.
The Willmore energy is unchanged by conformal mappings of the conformal 3-sphere. Its universal lower bound is the Willmore inequality, and its critical points are the Willmore surfaces.
