The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects

Abstract

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.