|
|
| Line 1: |
Line 1: |
| 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]].
| | Hello, my title is Andrew and my spouse doesn't like it at all. He functions as a bookkeeper. North Carolina is the place he loves most but now he is considering other options. What me and my family members adore is to climb but I'm thinking on starting some thing new.<br><br>My weblog; clairvoyance ([http://projects.topslabs.com/phpmelody_new/profile.php?u=LeSmalley linked website projects.topslabs.com]) |
| | |
| ==Definition==
| |
| Expressed symbolically, the Willmore energy of ''S'' is:
| |
| | |
| :<math> \mathcal{W} = \int_S H^2 \, dA - \int_S K \, dA</math>
| |
| | |
| 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
| |
| | |
| :<math> \int_S K \, dA = 2 \pi \chi(S), </math>
| |
| | |
| 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
| |
| :<math> \mathcal{W} = \int_S H^2 \, dA - 2 \pi \chi(S)</math>
| |
| | |
| An alternative, but equivalent, formula is
| |
| | |
| :<math> \mathcal{W} = {1 \over 4} \int_S (k_1 - k_2)^2 \, dA</math>
| |
| | |
| where <math>k_1</math> and <math>k_2</math> are the [[principal curvatures]] of the surface.
| |
| | |
| ===Properties===
| |
| The Willmore energy is always greater than or equal to zero. A round [[sphere]] has zero Willmore energy.
| |
| | |
| 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.
| |
| | |
| ==Critical points==
| |
| A basic problem in the [[calculus of variations]] is to find the [[critical point (mathematics)|critical points]] and minima of a functional.
| |
| | |
| For a given topological space, this is equivalent to finding the critical points of the function
| |
| :<math>\int_S H^2 \, dA </math>
| |
| since the Euler characteristic is constant.
| |
| | |
| One can find (local) minima for the Willmore energy by [[gradient descent]], which in this context is called [[#Willmore_flow|Willmore flow]].
| |
| | |
| 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>.
| |
| | |
| ==Willmore flow==
| |
| The '''Willmore flow''' is the [[geometric flow]] corresponding to the Willmore energy;
| |
| it is an <math>L^2</math>-[[gradient flow]].
| |
| | |
| :<math>e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A</math>
| |
| | |
| where ''H'' stands for the [[mean curvature]] of the [[manifold]] <math>\mathcal{M}</math>.
| |
| | |
| Flow lines satisfy the differential equation:
| |
| :<math> \partial_t x(t) = -\nabla \mathcal{W}[x(t)] \, </math>
| |
| where <math>x</math> is a point belonging to the surface.
| |
| | |
| This flow leads to an evolution problem in [[differential geometry]]: the surface <math>\mathcal{M}</math> is evolving
| |
| 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.
| |
| | |
| ==Applications==
| |
| * [[Cell membrane]]s tend to position themselves so as to minimize Willmore energy.
| |
| | |
| * Willmore energy is used in constructing a class of optimal [[sphere eversion]]s, the [[minimax eversion]]s.
| |
| | |
| ==See also==
| |
| * [[Willmore conjecture]]
| |
| | |
| ==References==
| |
| <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.
| |
| | |
| [[Category:Geometric flow]]
| |
| [[Category:Differential geometry]]
| |
| [[Category:Surfaces]]
| |
Hello, my title is Andrew and my spouse doesn't like it at all. He functions as a bookkeeper. North Carolina is the place he loves most but now he is considering other options. What me and my family members adore is to climb but I'm thinking on starting some thing new.
My weblog; clairvoyance (linked website projects.topslabs.com)