An index bound for smooth umbilic points

Abstract

We prove that the half-integer valued local index of an isolated umbilic point on a $C^{3+\alpha}$-smooth convex surface in Euclidean 3-space is less than two. The approach is to study the co-kernel of an associated Riemann-Hilbert boundary value problem. The link between the local and global is a semi-local technique that we term totally real blow-up. Topologically, given a real surface in a complex surface, the totally real blow-up is the connect sum of the real surface with an embedded real projective plane. We show that this increases the sum of the complex indices of the real surface by one, and hence cancels isolated hyperbolic complex points. This leads to the reduction of the local result to the global result (the non-existence of closed embedded Lagrangian surfaces with a single complex point), proving that the umbilic index for smooth surfaces is less than two. Comparison of our smooth result with that of Hans Hamburger in the real analytic case (stating that the index of an isolated umbilic point on a real analytic convex surface is less than or equal to one) suggests the existence of "exotic" umbilic points of index 3/2.