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 $D(\rho )\backslash \{0\}$ 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.