Willmore surface

In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore.

Expressed symbolically, the Willmore energy of S is:

where ${\displaystyle H}$ is the mean curvature, ${\displaystyle K}$ is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic ${\displaystyle \chi (S)}$ of the surface, so

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