Demystification of the Willmore integrand

The Willmore energy for a surface $S$ in Euclidean 3-space is defined as $\tilde{W}(S) = \int_S \mu^2\omega_S$ , where $\mu$ is the mean curvature of $S$ and $\omega_S$ its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form $\Omega_S = (\mu^2-K)\omega_S$ , where $K$ stands for the Gaussian curvature, is invariant under conformal transformations; its integral $W(S) = \int_S (\mu^2-K)\omega_S$ differs from the Willmore energy by the constant term, which equals $\int_SK\omega_S = 2\pi\chi(S)$ by Gauss–Bonnet theorem. Their derivation relies on the moving frame method, so the conformal invariance of this quantity is still mystifying.

We know, however, that a three-dimensional Riemannian manifold possesses another conformal invariant–namely, its LeBrun’s twistor space. As a smooth manifold, it is nothing more than the unit sphere bundle in its cotangent bundle, i. e. the bundle of oriented 2-planes. It carries a contact distribution, defined by $H_{\tau \susbet T_x} = \pi^*\tau$ ; and in three-dimensional case, it may be endowed with a complex structure operator. Indeed, the Levi-Civita connection splits the contact distribution into a sum of the vertical bundle $Ver_{\tau \subset T_x} \cong T_{\tau}ST^*_x$ and the horizontal bundle $Hor_{\tau \subset T_x} \cong \tau$ . Each of these bundles carries a natural complex structure operator; it turns out that its eigensubbundles in $H$ satisfy $\left[H^{1,0},H^{1,0}\right] \subset H^{1,0}$ and this structure is conformally invariant (though the Levi-Civita connection, and hence the horizontal subbundle, is not). LeBrun’s original construction did not involve arbitrary choices and exploited the complexified canonical 2-form on the cotangent bundle, restricted to the bundle of isotropic cones; however, for our purposes the outlined construction, which is probably due to Verbitsky, fits better.

Now let $X$ be a three-dimensional Riemannian manifold (which we are going deep in our hearts consider only up to conformal change of metric), and $S \subset X$ a cooriented surface. It inherits the conformal structure, which in dimension two is the same as the structure of a complex curve. The map $\ss \colon S \to ST^*X, s \mapsto T_sS$ is called the Gaussian map. It is horizontal, yet in general not holomorphic w. r. t. the LeBrun’s partial complex structure: it sends $v \in T_sS$ to $\nu_v(n)$ , where $n \in TX|_S$ is the unit normal vector field, and $\nabla$ the Levi-Civita connection. This map is nothing but the extrinsic curvature (known by a variety of names like Weingarten map or shape operator or second fundamental form), so its eigenvalues are the principal curvatures, and the Gaussian map is holomorphic at a point iff the principal curvatures are equal (such points are called umbilics).

The extrinsic curvature is of course not conformally invariant: though the composition of the Gaussian map $d\ss \colon T_sS \to T_{T_sS}ST^*X$ and the projection $d\pi \colon T_{T_sS}ST^*X \to T_sS$ is the identity map, in order to represent the tangent space to the Gaussian lift of a surface as a graph of a map $T_sS \cong Hor_{T_sS} \to Ver_{T_sS}$ , one must pick up the horizontal space. However, one can consider the antilinear part of the differential, $\bar{\partial}\ss \colon T^{1,0}_sS \to T^{0,1}_{T_sS}ST^*X$ , and its range is vertical, since the projection $T_{T_sS}ST^*X \to T_sS$ preserves the (1,0)-parts. This is much like the antilinear part of a holomorphic trivialization of a holomorphic vector bundle does not depend on the choice of the trivialization. So, for a surface $S$ in a conformal 3-manifold $X$ there exists a natural operator $\bar{\partial}\ss \colon T^{1,0}_s(S) \to T^{0,1}(ST^*_sX)$ . Notice that the fiber of the vertical bundle at the point $\tau \subset T_xX$ is isomorphic to $\mathrm{Hom}(\tau, T_x/\tau)$ , which is in our case $\mathrm{Hom}(T_sS, T_sX/T_sS)$ , so the operator $\bar{\partial}\ss$ lives naturally in $\mathrm{Hom}(T^{1,0}S\otimes T^{0,1}S, \nu(X/S))$ .

And if we choose a metric in the conformal class, the antilinear part of the shape operator writes down as $\begin{pmatrix}\frac{\kappa_1-\kappa_2}2 & 0 \\ 0 & -\frac{\kappa_1-\kappa_2}2\end{pmatrix}$ in the basis of principal directions, where $\kappa_i$ are the principal curvatures. In its determinant we easily recognize minus the Willmore integrand, which explains geometrically its conformal invariance not just for the round sphere case, but on any conformal 3-manifold.

References

* Robert L. Bryant. A duality theorem for Willmore surfaces, J. Differential Geom. 20(1) (1984): 23–53. * Claude R. LeBrun. Twistor $CR$ Manifolds and Three-Dimensional Conformal Geometry, Transactions of the American Mathematical Society 284, no. 2 (1984): 601–16.