Axially Symmetric Willmore Surfaces Determined by Quadratures

Abstract

The work is concerned with a special family of axially symmetric surfaces providing local extrema to the so-called Willmore functional, which assigns to each surface its total squared mean curvature. The components of the position vector of the profile curves of the regarded Willmore surfaces satisfy a system of first-order ordinary differential equations. The solutions of this system are expressed by quadratures in terms of the tangent angle and, in this way, the corresponding Willmore surfaces are determined.