An index formula for Loewner vector fields

Let $f$ be $C^2$ real-valued function defined near $0$ in $\Bbb R^2$, $ {\partial^2 f \over {\partial {\overline z}^2}} \neq 0$ for $z\ne 0$. Motivated by the Carathéodory conjecture in differential geometry, Loewner conjectured that the index at $0$ of the vector field given in complex notation by $ {\partial^2 f \over {\partial {\overline z}^2}} $ is at most two. In this paper we establish a formula that computes the index of these Loewner vector fields from data about the hessian of $f$.

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Published 1 January 2007