The Willmore inequality states that every smooth immersion of a closed surface 
into 3-dimensional Euclidean
space satisfies
where 
is the mean curvature, defined as the average of
the two principal curvatures, and 
is the area element. Equality
holds iff the image is a round sphere.
See also
Closed Surface,
Mean Curvature,
Sphere,
Willmore
Energy
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References
Pinkall, U. and Gross, O. "Willmore Surfaces." Ch. 13 in Differential
Geometry: From Elastic Curves to Willmore Surfaces. Cham, Switzerland: Birkhäuser,
2024. https://doi.org/10.1007/978-3-031-39838-4_13.Willmore,
T. J. Riemannian
Geometry. Oxford, England: Clarendon Press, 1993.
Cite this as:
Weisstein, Eric W. "Willmore Inequality."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WillmoreInequality.html
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