Math 2.0

Differential geometry Definition

Willmore flow

Builds on Willmore energyMean curvatureGaussian curvatureLaplace–Beltrami operator

Definition

Let Σ\Sigma be a closed oriented surface and let T>0T > 0. Let f ⁣:Σ×[0,T)→ℝ3f \colon \Sigma \times [0, T) \to \R^3 be a smooth family of immersions, and write ft=f(⋅,t)f_t = f(\cdot, t). For each tt, let ν\nu be the unit normal of ftf_t. Let H=12(κ1+κ2)H = \tfrac{1}{2}(\kappa_1 + \kappa_2) be the mean curvature of ftf_t with respect to ν\nu, so that the mean curvature vector is 2Hν2H\nu. Let KK be the Gaussian curvature of ftf_t. Let Δ\Delta be the Laplace–Beltrami operator of the metric induced by ftf_t. The family is a solution of the Willmore flow if ∂f∂t=−(ΔH+2H(H2−K))ν.\frac{\partial f}{\partial t} = -\left(\Delta H + 2H\left(H^2 - K\right)\right) \nu.

Remarks

The right side is minus the L2L^2 gradient of the Willmore energy W(ft)=∫ΣH2 dAW(f_t) = \int_\Sigma H^2 \, dA, so W(ft)W(f_t) does not increase. Example: the constant family of a round sphere of radius rr is a solution. Indeed H=±1/rH = \pm 1/r is constant, so ΔH=0\Delta H = 0, and H2−K=1/r2−1/r2=0H^2 - K = 1/r^2 - 1/r^2 = 0, so the right side vanishes. Non-example: the spheres of radius 1+t1 + t do not form a solution, since the right side vanishes for each of them but the surfaces move. A surface for which ΔH+2H(H2−K)=0\Delta H + 2H(H^2 - K) = 0 is a Willmore surface; these are exactly the stationary solutions. The equation is of fourth order and parabolic, so the maximum principle fails and convexity need not be preserved.

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Gaussian curvature, concept, no evidence levelGaussian curvatureconceptLaplace–Beltrami operator, concept, no evidence levelLaplace–Beltrami oper…conceptMean curvature, concept, no evidence levelMean curvatureconceptWillmore energy, concept, no evidence levelWillmore energyconceptWillmore flow, definition, no evidence levelWillmore flowdefinition
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  1. Gaussian curvature concept
  2. Laplace–Beltrami operator concept
  3. Mean curvature concept
  4. Willmore energy concept
  5. Willmore flow definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Willmore_energy
About this page
TagCMK92XDD0W
URIm2://CMK92XDD0W
Kinddefinition
WritingW0 stub
MSC53E40, 53A05
DefinesWillmore flow
UsesWillmore energy, Mean curvature, Gaussian curvature, Laplace–Beltrami operator
Changed2 November 2026

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