Analysis on Graphs and Fractals

Measurable Riemannian geometry on the Sierpinski gasket

J Kigami, Kyoto University, Japan

Abstract

We study the standard Dirichlet form and its energy measure, called the Kusuoka measure, on the Sierpinski gasket as a prototype of 'measurable Riemannian geometry'.  Kusuoka has shown that the standard DIrichlet form on the Sierpinski gasket can be expressed by an integration of measurable counterpart of  Riemannian metric, gradient operator under the energy measure. The shortest path metric on the harmonic Sierpinski gasket is shown to be the geodesic distance associated with the 'measurable Riemannian structure'. The Kusuoka measure is shown to have the volume doubling property with respect to the Euclidean distance and also to the geodesic distance. Li-Yau type Gaussian off-diagonal heat kernel estimate is established for the heat kernel associated with the Kusuoka measure.