On a Conjecture of Carathéodoryr Analyticity Versus Smoothness

Abstract

We show that, under mild nonflatness conditions, for any r ≥ 3 and any C ^r -immersion of a surface into R³ with an isolated urnblhc point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Caratheodory's Conjecture on umbilics is discussed.