Radially Symmetric Solutions To The Graphic Willmore Surface Equation

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 $D(ρ)\backslash\{0\}$ and is in $C^1(D(ρ))$, we show that there exist a constant $λ$ and a function $β$ in $C^0(D(ρ))$ such that $u''(r) =\fracλ{2}\log r+β(r)$; moreover, the graph of $u$ is contained in a graphical region of an inverted catenoid which is uniquely determined by $λ$ and $β(0)$. 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.

We extend our previous study of smooth radial solutions to solutions with a singularity at 0. A new section is added