Radially Symmetric Solutions To The Graphic Willmore Surface Equation

TL;DR

This paper classifies radially symmetric solutions to the graphic Willmore surface equation, showing they are either constant or half spheres, and analyzes solutions on punctured disks relating them to inverted catenoids.

Contribution

It provides a complete classification of smooth radially symmetric solutions and describes their behavior near singularities, linking solutions to geometric objects like catenoids.

Findings

Radially symmetric entire solutions are flat or half spheres.

Solutions on punctured disks relate to inverted catenoids.

Radial solutions extend to the disk when mean curvature is square integrable.

Abstract

We show that a smooth radially symmetric solution $u$ to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ must be flat. When $u$ is a smooth radial solution over a punctured disk $D(\rho)\backslash\{0\}$ and is in $C^1(D(\rho))$, we show that there exist a constant $\lambda$ and a function $\beta$ in $C^0(D(\rho))$ such that $u''(r) =\frac{\lambda}{2}\log r+\beta(r)$; moreover, the graph of $u$ is contained in a graphical region of an inverted catenoid which is uniquely determined by $\lambda$ and $\beta(0)$. It is also shown that a radial solution on the punctured disk extends to a $C^1$ function on the disk when the mean curvature is square integrable.