We define a new notion of curvature, called Loewner curvature, so-named because it captures key behavior of the trace curve of the Loewner differential equation. The Loewner curvature is defined for (nice enough) curves that begin at a marked boundary point of a Jordan domain and grow towards a second marked boundary point. We show that if this curvature is small, then the curve must remain a simple curve. This is joint work with Steffen Rohde.
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