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.