The Willmore equation is the Euler-Lagrange differential equation for the Willmore energy.
For an immersion in 3-dimensional Euclidean
space, with mean curvature 
, it is
where 
is the Laplacian of the induced metric
and 
is the Gaussian curvature. The equation is
invariant under conformal mappings, and its
solutions are the Willmore surfaces.
See also
Euler-Lagrange Differential Equation,
Willmore Energy,
Willmore Surface
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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 Equation."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WillmoreEquation.html
Subject classifications
