On a conjecture of Carathéodory: analyticity versus smoothness

Abstract

We show that, under mild nonflatness conditions, for any $r\ge 3$ and any $C^r$-immersion of a surface into $\R^3$ with an isolated umbilic point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Carathéodory's Conjecture on umbilics is discussed.