Mathematical Research Letters

Volume 14, Issue 5, September 2007 pp. 865-873.

An index formula for Loewner vector fields

Authors: Frederico Xavier
Author institution: University of Notre Dame

Summary: 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 Carath\'eodory 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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