# Willmore energy

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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 closed surface 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. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

## Definition

Expressed symbolically, the Willmore energy of S is:

${\mathcal {W}}=\int _{S}H^{2}\,dA-\int _{S}K\,dA$ where $H$ is the mean curvature, $K$ 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 $\chi (S)$ of the surface, so

In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. In differential geometry, the Gaussian curvature or Gauss curvatureΚ of a surface at a point is the product of the principal curvatures, κ1 and κ2, at the given point: The Gauss–Bonnet theorem, or Gauss–Bonnet formula, is an important statement in differential geometry about surfaces which connects their geometry to their topology. It is named after Carl Friedrich Gauss, who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

$\int _{S}K\,dA=2\pi \chi (S),$ which is a topological invariant and thus independent of the particular embedding in $\mathbb {R} ^{3}$ that was chosen. Thus the Willmore energy can be expressed as

In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

${\mathcal {W}}=\int _{S}H^{2}\,dA-2\pi \chi (S)$ An alternative, but equivalent, formula is

${\mathcal {W}}={1 \over 4}\int _{S}(k_{1}-k_{2})^{2}\,dA$ where $k_{1}$ and $k_{2}$ 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.

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

## Critical points

A basic problem in the calculus of variations is to find the critical points and minima of a functional.

For a given topological space, this is equivalent to finding the critical points of the function

$\int _{S}H^{2}\,dA$ 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.

For embeddings of the sphere in 3-space, the critical points have been classified:  they are all conformal transforms of minimal surfaces, the round sphere is the minimum, and all other critical values are integers greater than or equal to 4$\pi$ . They are called Willmore surfaces.

## Willmore flow

The Willmore flow is the geometric flow corresponding to the Willmore energy; it is an $L^{2}$ -gradient flow.

$e[{\mathcal {M}}]={\frac {1}{2}}\int _{\mathcal {M}}H^{2}\,\mathrm {d} A$ where H stands for the mean curvature of the manifold ${\mathcal {M}}$ .

Flow lines satisfy the differential equation:

$\partial _{t}x(t)=-\nabla {\mathcal {W}}[x(t)]\,$ where $x$ is a point belonging to the surface.

This flow leads to an evolution problem in differential geometry: the surface ${\mathcal {M}}$ is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion it is a fourth-order flow, since the variation of the energy contains fourth derivatives.