Willmore energy
Encyclopedia
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 mean curvature
squared minus the Gaussian curvature
. It is named after the English geometer Tom Willmore
.
where
is the mean curvature
,
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
of the surface, so
which is a topological invariant
and thus independent of the particular embedding in
that was chosen. Thus the Willmore energy can be expressed as
An alternative, but equivalent, formula is
where
and 
are the principal curvatures of the surface.
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.
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
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 surface
s, the round sphere is the minimum, and all other critical values are integers greater than or equal to 4.
corresponding to the Willmore energy
;
it is an
-gradient flow.
where H stands for the mean curvature
of the manifold
.
Flow lines satisfy the differential equation:
where
is a point belonging to the surface.
This flow leads to an evolution problem in differential geometry: the surface
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.
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
deviates from a round sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. Mathematically, the Willmore energy of a smooth closed surface embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in three-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is defined to be the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the mean curvature
Mean curvature
In mathematics, the mean curvature H of a surface S 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....
squared minus the Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
. It is named after the English geometer Tom Willmore
Tom Willmore
Thomas James Willmore was an English geometer. He is best known for his work on Riemannian 3-space and harmonic spaces.Willmore studied at King's College London...
.
Definition
Expressed symbolically, the Willmore energy of S is:where
is the mean curvatureMean curvature
In mathematics, the mean curvature H of a surface S 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....
,
is the Gaussian curvatureGaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
, 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
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of the surface, sowhich is a topological invariant
Topological property
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...
and thus independent of the particular embedding in
that was chosen. Thus the Willmore energy can be expressed asAn alternative, but equivalent, formula is
where
and are the principal curvatures of the surface.Properties
The Willmore energy is always greater or equal to zero. A sphereSphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
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
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, and one can vary the embedding of a surface, while leaving it topologically unaltered.
Critical points
A basic problem in the calculus of variationsCalculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
is to find the critical points
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
and minima of a functional.
For a given topological space, this is equivalent to finding the critical points of the function
since the Euler characteristic is constant.
One can find (local) minima for the Willmore energy by gradient descent
Gradient descent
Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point...
, 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 surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s, the round sphere is the minimum, and all other critical values are integers greater than or equal to 4.
Willmore flow
The Willmore flow is the geometric flowGeometric flow
In mathematics, specifically differential geometry, a geometric flow is the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature...
corresponding to the Willmore energy
Willmore energy
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 mean curvature squared...
;
it is an
-gradient flow.where H stands for the mean curvature
Mean curvature
In mathematics, the mean curvature H of a surface S 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....
of the manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
.Flow lines satisfy the differential equation:
where
is a point belonging to the surface.This flow leads to an evolution problem in differential geometry: the surface
is evolvingin 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 membraneCell membraneThe cell membrane or plasma membrane is a biological membrane that separates the interior of all cells from the outside environment. The cell membrane is selectively permeable to ions and organic molecules and controls the movement of substances in and out of cells. It basically protects the cell...
s tend to position themselves so as to minimize Willmore energyWillmore energyIn 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 mean curvature squared...
.
- Willmore energy is used in constructing a class of optimal sphere eversions, the minimax eversionMinimax eversionIn geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models.It is a variational method, and consists of special homotopies ; contrast with Thurston's corrugations, which are generic.The original method of half-way models was not optimal: the regular...
s.