Harmonic embeddings of the Stretched Sierpinski Gasket

Abstract

P. Alonso-Ruiz, U. Freiberg and J. Kigami have defined a large family of resistance forms on the Stretched Sierpinski Gasket G. In the present paper we introduce a system of coordinates on G (technically, an embedding of G into R 2 ) such that• these forms are defined on C 1 (R 2 , R) and • all affine functions are harmonic for them. We do this adapting a standard method from the Harmonic Sierpinski Gasket: we start finding a sequence G l of pre-fractals such that all affine functions are harmonic on G l . After showing that this property is inherited by the stretched harmonic gasket G, we use the formula for the Laplacian of a composition to prove that, for a natural measure μ on G, C 2 (R 2 , R) ⊂ D(Δ) and Teplyaev's formula for the Laplacian of C 2 functions holds. Lastly, we use the expression for Δu to show that the form we have found is closable in L 2 (G, μ).