The Willmore energy for a surface in Euclidean 3-space is defined as , where is the mean curvature of and 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 , where stands for the Gaussian curvature, is invariant under conformal transformations; its integral differs from the Willmore energy by the constant term, which equals 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 ; 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 and the horizontal bundle . Each of these bundles carries a natural complex structure operator; it turns out that its eigensubbundles in satisfy 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 be a three-dimensional Riemannian manifold (which we are going deep in our hearts consider only up to conformal change of metric), and a cooriented surface. It inherits the conformal structure, which in dimension two is the same as the structure of a complex curve. The map is called the Gaussian map. It is horizontal, yet in general not holomorphic w. r. t. the LeBrun’s partial complex structure: it sends to , where is the unit normal vector field, and 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 and the projection is the identity map, in order to represent the tangent space to the Gaussian lift of a surface as a graph of a map , one must pick up the horizontal space. However, one can consider the antilinear part of the differential, , and its range is vertical, since the projection 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 in a conformal 3-manifold there exists a natural operator . Notice that the fiber of the vertical bundle at the point is isomorphic to , which is in our case , so the operator lives naturally in .
And if we choose a metric in the conformal class, the antilinear part of the shape operator writes down as in the basis of principal directions, where 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.
* Robert L. Bryant. A duality theorem for Willmore surfaces, J. Differential Geom. 20(1) (1984): 23–53.
* Claude R. LeBrun. Twistor Manifolds and Three-Dimensional Conformal Geometry, Transactions of the American Mathematical Society 284, no. 2 (1984): 601–16.