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# Mathematical Research Letters

## Volume 14 (2007)

### Number 5

### An index formula for Loewner vector fields

Pages: 865 – 873

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n5.a13

#### Author

#### Abstract

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 $f$.