IUMJ

Title: Curvature densities of self-similar sets

Authors: Jan Rataj and Martina Zahle

Issue: Volume 61 (2012), Issue 4, 1425-1449

Abstract:

For a large class of self-similar sets $F$ in $\mathbb{R}^d$, analogues of the higher-order mean curvatures of differentiable submanifolds are introduced---in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated fractal curvature measures, which are all multiples of the corresponding Hausdorff measures on $F$, due to its self-similarity. This local approach based on ergodic theory for an associated dynamical system enables us to extend former total curvature results.