Willmore canal surfaces in Euclidean space

Abstract

We study envelopes of 1-parameter families of spheres (including planes) in Euclidean space which are critical points of the Willmore functional (Willmore canal surfaces). We prove that Willmore canal surfaces are isothermic surfaces and hence conformally equivalent to surfaces of revolution, cones or cylinders. We provide explicit formulae for all solution surfaces. In the generic case the formulae involve Weierstrass's elliptic functions. There are two exceptional cases which can be integrated by using elementary functions only, namely the catenoid and the stereographic projection of the minimal Clifford torus in S3. To obtain the solution surfaces we explicitly integrate the linear differential system defining the Willmore canal surfaces.