Harmonic coordinates on finitely connected fractafolds
Alexander Teplyaev · arXiv (Cornell University) · 2005
Abstract. We define finitely connected fractafolds, which are generalizations of p.c.f. self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. Any self-similarity is not assumed, and countably infinite ramification is allowed. We prove that if a fractafold has a resistance form in the sense of Kigami that satisfies certain assumptions, then there exists a weak Riemannian metric, defined almost everywhere, such that the energy can be expressed as the integral of the norm of a weak gradient with respect to an energy measure. This generalizes earlier results by Kusuoka and the author. Furthermore, we prove that if the fractafold can be homeomorphically represented in harmonic coordinates, then the weak gradient can be replaced by the usual gradient for smooth functions, which generalizes an earlier result by Kigami. We also prove a simple formula for the energy measure