Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets

TL;DR

This paper establishes a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gaskets and derives Gaussian heat kernel bounds, using geodesics characterized as de Rham curves with known regularity.

Contribution

It introduces a new approach to analyze higher-dimensional harmonic Sierpinski gaskets by proving geodesics are de Rham curves, enabling Gaussian heat kernel estimates.

Findings

Existence of a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gaskets.

Gaussian heat kernel bounds in the geodesic metric.

Geodesics are shown to be de Rham curves with extensive regularity theory.

Abstract

We prove existence of a measurable Riemannian structure on higher-dimensional harmonic Sierpinski gasket fractals and deduce Gaussian heat kernel bounds in the geodesic metric. Our proof differs from that given by Kigami for the usual Sierpinski gasket in that we show the geodesics are de Rham curves, for which there is an extensive regularity theory.