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| 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 manifold|smooth]] [[closed surface]] [[Embedding|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]].
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| ==Definition==
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| Expressed symbolically, the Willmore energy of ''S'' is:
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| :<math> \mathcal{W} = \int_S H^2 \, dA - \int_S K \, dA</math>
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| where <math>H</math> is the [[mean curvature]], <math>K</math> 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]] <math>\chi(S)</math> of the surface, so
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| :<math> \int_S K \, dA = 2 \pi \chi(S), </math>
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| which is a [[topological property|topological invariant]] and thus independent of the particular embedding in <math>\mathbb{R}^3</math> that was chosen. Thus the Willmore energy can be expressed as
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| :<math> \mathcal{W} = \int_S H^2 \, dA - 2 \pi \chi(S)</math>
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| An alternative, but equivalent, formula is
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| :<math> \mathcal{W} = {1 \over 4} \int_S (k_1 - k_2)^2 \, dA</math>
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| where <math>k_1</math> and <math>k_2</math> are the [[principal curvatures]] of the surface.
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| ===Properties===
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| The Willmore energy is always greater than or equal to zero. A round [[sphere]] has zero Willmore energy.
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| The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the [[calculus of variations]], and one can vary the embedding of a surface, while leaving it topologically unaltered.
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| ==Critical points==
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| A basic problem in the [[calculus of variations]] is to find the [[critical point (mathematics)|critical points]] and minima of a functional.
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| For a given topological space, this is equivalent to finding the critical points of the function
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| :<math>\int_S H^2 \, dA </math>
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| since the Euler characteristic is constant.
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| One can find (local) minima for the Willmore energy by [[gradient descent]], which in this context is called [[#Willmore_flow|Willmore flow]].
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| For embeddings of the sphere in 3-space, the critical points have been classified:<ref>Robert Bryant. A duality theorem for Willmore surfaces. J. Differential Geometry 20(1984), 23–53.</ref> they are all [[conformal transform]]s of [[minimal surface]]s, the round sphere is the minimum, and all other critical values are integers greater than or equal to 4<math>\pi</math>.
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| ==Willmore flow==
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| The '''Willmore flow''' is the [[geometric flow]] corresponding to the Willmore energy;
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| it is an <math>L^2</math>-[[gradient flow]].
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| :<math>e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A</math> | |
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| where ''H'' stands for the [[mean curvature]] of the [[manifold]] <math>\mathcal{M}</math>.
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| Flow lines satisfy the differential equation:
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| :<math> \partial_t x(t) = -\nabla \mathcal{W}[x(t)] \, </math>
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| where <math>x</math> is a point belonging to the surface.
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| This flow leads to an evolution problem in [[differential geometry]]: the surface <math>\mathcal{M}</math> is evolving
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| in time to follow variations of steepest descent of the energy. Like [[surface diffusion (mathematics)]] it is a fourth-order | |
| flow, since the variation of the energy contains fourth derivatives.
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| ==Applications==
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| * [[Cell membrane]]s tend to position themselves so as to minimize Willmore energy.
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| * Willmore energy is used in constructing a class of optimal [[sphere eversion]]s, the [[minimax eversion]]s.
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| ==See also==
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| * [[Willmore conjecture]]
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| ==References==
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| <references/> | |
| * Thomas J. Willmore. A survey on Willmore immersions. In Geometry and Topology of Submanifolds, IV (Leuven, 1991), pp 11–16. World Sci. Pub., 1992.
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| [[Category:Geometric flow]]
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| [[Category:Differential geometry]]
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| [[Category:Surfaces]]
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