Willmore Surfaces

Abstract

The analog for a surface $f\colon M\to \mathbb {R}^3$ of the bending energy $\int _a^b\kappa ^2\,ds$ is the Willmore functional$W(f)=\int _M H^2\det$.

The analog for a surface $f\colon M\to \mathbb {R}^3$ of the bending energy $\int _a^b\kappa ^2\,ds$ is the Willmore functional$W(f)=\int _M H^2\det$. There are several versions of the Willmore functional, all of which are equivalent for the purposes of Variational Calculus. One of these versions is unchanged if we transform the surface by inversion in a sphere. The analogs of elastic curves are called Willmore Surfaces.

1 The Willmore Functional

In the context of curves $\gamma \colon [a,b]\to \mathbb {R}^2$ we studied in detail the total squared curvature $\int \kappa ^2 ds$ (notation from the end of Sect. 7.2). What is the analog of this energy in the context of surfaces?

One might say that $\kappa =0$ characterizes straight lines, which minimize length among all curves with the same end points. So $\int _{[a,b]}\kappa ^2 \,ds$ measures the deviation from being length-minimizing. The analog of length-minimizing curves are area-minimizing surfaces, i.e. minimal surfaces, surfaces with mean curvature $H=0$. So a natural analog of $\int _{[a,b]}\kappa ^2 \,ds$ can be defined as follows:

Definition 13.1

If $f\colon M\to \mathbb {R}^3$ is a surface, then

$$\displaystyle \begin{aligned} W(f):=\int_M H^2\det\end{aligned}$$

is called the Willmore functional of f.

Surfaces f that are critical points of the Willmore functional are characterized by the property that they are “as minimal as possible”, given that they are held fixed near the boundary of M.

Alternatively, one might say that $\kappa =0$ only happens for straight line segments, so for surfaces we want to measure the deviation of being planar. Parametrizations of pieces of the plane are characterized by the fact that both principal curvatures vanish, so we want to measure the deviation of both $\kappa _1$ and $\kappa _2$ (not just their average H) from being zero. This reasoning leads to a different analog for $\int _{[a,b]}\kappa ^2 \,ds$:

Definition 13.2

If $f\colon M\to \mathbb {R}^3$ is a surface, then

$$\displaystyle \begin{aligned} E(f):=\frac{1}{4}\int_M (\kappa_1^2+\kappa_2^2)\det=\int_M \left(H^2-\frac{K}{2}\right)\det\end{aligned}$$

is called the bending energy of f.

Surfaces f that are critical points of the bending energy are characterized by the property that they are “as planar as possible”, given that they are held fixed near the boundary.

Finally, one might formulate a different wish and ask for surfaces that are “as round as possible” which means they are “as spherical as possible”. In view of the Umbillic Point Theorem 8.12 this motivates the following definition:

Definition 13.3

If $f\colon M\to \mathbb {R}^3$ is a surface, then

$$\displaystyle \begin{aligned} \widetilde{W}(f):=\frac{1}{4}\int_M (\kappa_1-\kappa_2)^2 \det=\int_M (H^2-K)\det\end{aligned}$$

is called the conformally invariant Willmore functional of f.

Note that the integrands in all three of the above energies differ only by a term proportional to $K\det$, so the Gauss-Bonnet Theorem 10.6 tells us that for the purposes of Variational Calculus, (cf. Definition A.4) all three energies are equivalent to a large extent:

Theorem 13.4

Let$f,\tilde {f}\colon M\to \mathbb {R}^3$be two surfaces such that$\tilde {f}(p)=f(p)$for p outside of some compact set contained in the interior of M. Then

$$\displaystyle \begin{aligned} W(\tilde{f})-W(f)=E(\tilde{f})-E(f)=\widetilde{W}(\tilde{f})-\widetilde{W}(f).\end{aligned}$$

For surfaces that close up we see that the difference between the three functionals only depends on the genus:

Theorem 13.5

Let$f\colon M\to \mathbb {R}^3$be a surface that closes up with genus g. Then

$$\displaystyle \begin{aligned} E(f)&=W(f)+2\pi(g-1) \\ \widetilde{W}(f)&=W(f) + 4\pi (g-1).\end{aligned} $$

Theorem 13.6

The estimates below are sharp, i.e. in each case there is a surface that closes up with the prescribed genus and which realizes the lower bound:

• (i) If M is connected and a surface$f\colon M\to \mathbb {R}^3$closes up with genus 0, then

  $$\displaystyle \begin{aligned} W(f)\geq 4\pi.\end{aligned}$$

• (ii) If M is connected and a surface$f\colon M\to \mathbb {R}^3$closes up with genus$\frac {1}{2}$, then

  $$\displaystyle \begin{aligned} W(f)\geq 12\pi.\end{aligned}$$

• (iii) If M is connected and a surface$f\colon M\to \mathbb {R}^3$closes up with genus 1, then

  $$\displaystyle \begin{aligned} W(f)\geq 2\pi^2.\end{aligned}$$

We will not prove this theorem. Part (i) of Theorem 13.6 was proved by Tom Willmore in [46] in 1965. The minimum is attained for a round sphere. Part (ii) was proved by Rob Kusner in [21] where he also proved that the Boy surface shown on the right of Fig. 13.1 realizes the minimum $12\pi$. The two surfaces on the right of Fig. 13.2 are Lawson surfaces which were found by Blaine Lawson [23] and are possible candidates for minimizing the Willmore functional among all surfaces with genus $g=2$ and $g=3$ respectively [18].

Fig. 13.1

A Boy surface with the minimal possible Willmore functional $12\pi$

Fig. 13.2

The Torus on the left has Willmore functional $2\pi ^2$, which is optimal among surfaces with genus $g=1$. The two surfaces on the right are possible candidates for minimizing the Willmore functional among all surfaces with genus $g=2$ and $g=3$ respectively

In the paper already mentioned above, Willmore also formulated (iii) as a conjecture and demonstrated that the value $2\pi ^2$ is realized by the torus obtained by rotating a circle of radius one around an axis in such a way that its center has distance $\sqrt {2}$ from the axis (Fig. 13.2, left). This Willmore conjecture remained a famous open problem in Differential Geometry for a long time, until in 2012 Fernando Marques and André Neves proved the conjecture [26].

Remark 13.7

The question of critical points of the Willmore functional acquired greater importance starting from the 1960s, initiated by T. Willmore and his paper [46]. It was later found that parts of the theory were already known to Wilhelm Blaschke [6] and his student Gerhard Thomsen in the 1920s [38]. For an historic overview of contributions which were made to the problem see the last chapter of [47], or [27] for a more recent survey.

2 Variation of the Willmore Functional

According to the discussion in Sect. 13.1, the Willmore functional $\mathcal {W}$ has alternative versions which measure how “non-flat” or how “not round” a surface is. It was also explained that for the purposes of Variational Calculus all these different versions of the Willmore functional are equivalent. Being a critical point of the Willmore functional (which version we take does not matter) means that the surface (at least locally) is “optimally round”. It also means that the total amount of curvature of the surface cannot be decreased by modifying f only in a small neighborhood of a given point, while leaving the rest of the surface unchanged.

Definition 13.8

A surface $f\colon M\to \mathbb {R}^3$ is called a Willmore surface if it is a critical point of the Willmore functional $\mathcal {W}$.

Let us first compute for $\mathcal {W}$ the rate of change under a general variation, not necessarily with support in the interior:

Theorem 13.9 (First Variation Formula for the Willmore Functional)

Let$f\colon M\to \mathbb {R}^3$be a surface with unit normal N and with binormal field B along the boundary$\partial M$. Let$t\mapsto f_t$be a variation of f with variational vector field

where$\phi \in C^\infty (M)$and$Z\in \Gamma (TM)$. Then

$$\displaystyle \begin{aligned} \left.\frac{d}{dt}\right|{}_{t=0}\mathcal{W}(f_t) &= -\int_M \phi\left(\Delta H +2H(H^2-K)\right)\det \\ &\quad + \int_{\partial M}\left\langle B,H^2 Z-H\,\mathrm{grad}\,\phi+\phi \,\mathrm{grad}\,H\right\rangle\,ds.\end{aligned} $$

Proof

Using Theorem 12.18 as well as the notation $G:=\mathrm {grad}\,\phi$ borrowed from there we obtain

$$\displaystyle \begin{aligned} {\left(H^2\det\right)\,}^{\mbox{\bf \hspace{-0.35ex}.}} &= \left(2H\left(d_Z H -\frac{1}{2} \mathrm{div} \,G -\phi(2H^2-K)\right)+H^2(2H\phi+\mathrm{div}\,Z)\right)\det\\ &=\left(2H\langle \mathrm{grad}\,H,Z\rangle -H\,\mathrm{div}\,G \,-\, 2\phi H (H^2-K) +H^2\mathrm{div}\,Z\right)\det \\ &=\left(\mathrm{div}(H^2Z)-H\,\mathrm{div}\,G -2\phi H(H^2-K)\right)\det \\ &=\left(\mathrm{div}(H^2Z-HG)+\langle \mathrm{grad}\,H,\mathrm{grad}\,\phi\rangle-2\phi H(H^2-K)\right)\det \\ &=\left(\mathrm{div}(H^2Z-HG)+\mathrm{div}(\phi\,\mathrm{grad}\,H)-\phi \Delta H-2\phi H(H^2-K)\right)\det \\ &=\left(\mathrm{div}\left(H^2Z-HG+\phi \,\mathrm{grad}\,H\right)-\phi\left(\Delta H +2H(H^2-K)\right)\right)\det.\end{aligned} $$

Together with the Divergence Theorem 12.7, this proves our claim. □

As an immediate consequence, we obtain (cf. [38])

Theorem 13.10

A surface$f\colon M\to \mathbb {R}^3$is a Willmore surface if and only if

$$\displaystyle \begin{aligned} \Delta H +2H(H^2-K)=0.\end{aligned}$$

Round spheres are Willmore, because for them all points are umbilic points (so $H^2-K=0$) and H is constant (so $\Delta H=0$). Moreover, all surfaces with $H=0$ (minimal surfaces) are Willmore. Here is another example:

Example 13.11

Take a unit speed curve $\gamma \colon [0,L]\to \mathbb {R}^2\subset \mathbb {R}^3$, where $\mathbb {R}^2$ is realized as those points in $\mathbb {R}^3$ where the last coordinate is zero. Now for a compact domain with smooth boundary $M\subset [0,L]\times \mathbb {R}$ define the cylinder$f\colon M\to \mathbb {R}^3$ over $\gamma$ by

$$\displaystyle \begin{aligned} f(u,v)=\gamma(u)+v\,{\mathbf{e}}_3.\end{aligned}$$

It is easy to check that the Levi-Civita connection of f is given by $\nabla U=\nabla V=0$, the Gaussian curvature K of f vanishes and the mean curvature H of f satisfies

$$\displaystyle \begin{aligned} H(u,v)&=\frac{\kappa(u)}{2} \\ (\mathrm{grad}\,H)(u,v)&=\frac{\kappa'(u)}{2}U(u,v) \\ (\Delta H)(u,v)&= \frac{\kappa^{\prime\prime}(u)}{2}.\end{aligned} $$

This means that the cylinder f over $\gamma$ is Willmore if and only if $\gamma$ is freely elastic, i.e.

$$\displaystyle \begin{aligned} \kappa^{\prime\prime}+\frac{\kappa^3}{2}=0.\end{aligned}$$

The cylinder over a freely elastic curve is seen in Fig. 13.3.

Fig. 13.3

The cylinder over a free elastic plane curve is a Willmore surface

There are many other ways to construct Willmore surfaces, most of which are beyond the scope of this book. The surface in Fig. 13.4 is from the 2019 paper [7].

Fig. 13.4

Another Willmore surface

3 Willmore Functional Under Inversions

For a surface $f\colon M\to \mathbb {R}^3$, the Willmore functional

$$\displaystyle \begin{aligned} \mathcal{W}(f)=\int_M H^2 \,\det\end{aligned}$$

is clearly unchanged if we postcompose f by an isometry $g\colon \mathbb {R}^3 \to \mathbb {R}^3$. It is also invariant under scaling: For $\lambda \neq 0$ the surface $\tilde {f}=\lambda f$ has the same Willmore functional. This is because, under such a scaling, $\det$ acquires a factor of $\lambda ^2$ while H gets a factor of $\frac {1}{\lambda }$. As its name indicates, if we consider the Möbius-invariant Willmore functional

$$\displaystyle \begin{aligned} \widetilde{\mathcal{W}}(f)=\int_M (H^2-K)\det\end{aligned}$$

a similar statement is true for a more general class of transformations, that can be written as compositions of isometries, scalings and inversions in spheres, the so-called Möbius transformations (Fig. 13.5).

Fig. 13.5

The images under an inversion of the surfaces in Figs. 13.3 and 6.8 respectively are also Willmore surfaces

Let $f\colon M\to \mathbb {R}^3$ be a surface such that the origin $\mathbf {o}$ of $\mathbb {R}^3$ is not in the image of f. Then we can postcompose f with the so-called inversion in the unit sphere

$$\displaystyle \begin{aligned} g\colon\mathbb{R}^3 \setminus\{\mathbf{o}\} \to \mathbb{R}^3,\ g(\mathbf{p})=\frac{\mathbf{p}}{\langle \mathbf{p},\mathbf{p}\rangle}\end{aligned} $$

and obtain a new surface

$$\displaystyle \begin{aligned} \tilde{f}\colon M \to \mathbb{R}^3,\ \tilde{f} = \frac{f}{\langle f,f\rangle}.\end{aligned} $$

Computing the derivative of $\tilde {f}$ is straightforward and yields

$$\displaystyle \begin{aligned} d\tilde{f}&=\frac{df}{\langle f,f\rangle} -2 \frac{\langle df,f\rangle f}{\langle f,f\rangle^2} \\ &=\frac{1}{\langle f,f\rangle} R\,df\end{aligned} $$

where for each $p\in M$ the orthogonal $(3\times 3)$-matrix $R(p)\in O(3)$ acts on $\mathbf {v}\in \mathbb {R}^3$ as

$$\displaystyle \begin{aligned} R(p)\mathbf{v} = \mathbf{v}-2\frac{\langle f(p),\mathbf{v}\rangle}{\langle f(p),f(p)\rangle}f(p).\end{aligned}$$

For each $p\in M$ the matrix $R(p)$ is a reflection and hence orientation-reversing. The sign of the unit normal depends on orientation, which is why the unit normal field of $\tilde {f}$ is given by

$$\displaystyle \begin{aligned} \tilde{N}=-RN =\frac{2\langle N,f\rangle}{\langle f,f\rangle}f-N.\end{aligned}$$

Theorem 13.12

In the situation above, the induced metric$\langle \,,\rangle ^\sim$, the area form$\widetilde {\det }$and the shape operator$\tilde {A}$of$\tilde {f}$are given by

$$\displaystyle \begin{aligned} \langle \,,\rangle^\sim &= \frac{1}{\langle f,f\rangle^2}\langle \,,\rangle \\ \widetilde{\det} &= \frac{1}{\langle f,f\rangle^2} \det \\ \tilde{A}&=-\langle f,f\rangle A+2\langle N,f\rangle I.\end{aligned} $$

Proof

The first two formulas follow directly from our calculations above. The third follows from

$$\displaystyle \begin{aligned} d\tilde{N}&=\left(\frac{2\langle dN,f\rangle}{\langle f,f\rangle} -\frac{4\langle N,f\rangle \langle df,f\rangle}{\langle f,f\rangle^2}\right)f+\frac{2\langle N,f\rangle}{\langle f,f\rangle}df-dN \\&=2\langle N,f\rangle d\tilde{f} -\langle f,f\rangle d\tilde{f}\circ A.\end{aligned} $$

□

Theorem 13.13

If$\tilde {f}$arises from f by inversion in the unit sphere, then

$$\displaystyle \begin{aligned} \widetilde{\mathcal{W}}(\tilde{f})=\widetilde{\mathcal{W}}(f).\end{aligned}$$

Proof

By Theorem 13.12, the principal curvatures $\tilde {\kappa }_1,\tilde {\kappa }_2$ of $\tilde {f}$ satisfy

$$\displaystyle \begin{aligned} \tilde{\kappa}_2-\tilde{\kappa}_1=-\langle f,f\rangle(\kappa_2-\kappa_1).\end{aligned}$$

As a consequence,

$$\displaystyle \begin{aligned} (\tilde{H}^2-\tilde{K})\widetilde{\det}=\frac{1}{4}(\tilde{\kappa}_2-\tilde{\kappa_1})^2\,\widetilde{\det}=\frac{1}{4}(\kappa_2-\kappa_1)^2\det =(H^2-K)\det.\end{aligned}$$

□

Theorem 13.14

If f is a Willmore surface, then so is its image$\tilde {f}$under inversion in the unit sphere.

Proof

By Theorem 13.4, $\widetilde {\mathcal {W}}$ has the same critical points as $\mathcal {W}$ and by Theorem 13.13 inversion in the unit sphere maps critical points of $\widetilde {\mathcal {W}}$ to critical points of $\widetilde {\mathcal {W}}$. □

The surface on the right of Fig. 13.6 shows the image under an inversion of a minimal surface already known to Euler (shown on the left), the so-called catenoid$f\colon M\to \mathbb {R}^3$ given by

$$\displaystyle \begin{aligned} f(u,v)=\left(\begin{array}{c}\frac{1+u^2+v^2}{u^2+v^2}u \\ \frac{1+u^2+v^2}{u^2+v^2}v \\ \log(u^2+v^2)\end{array}\right).\end{aligned}$$

Fig. 13.6

A catenoid(left) and its image under a sphere inversion (right)

So, even more Willmore surfaces can be obtained by inverting surfaces which we have already encountered (see Fig. 13.5).