Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket

Conformal Energy, Conformal Laplacian, and Energy Measures on the Sierpinski Gasket

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy Eφ and conformal Laplacian ∆φ for a given conformal factor φ, based on the corresponding notions in Riemannian geometry in dimension n 6= 2. We derive a differential equation that describes the dependence of the effective resistances of Eφ on φ. We show that the spectrum of∆φ (Dirichlet or Neumann) has si...

Energy and Laplacian on the Sierpiński Gasket

This is an expository paper which includes several topics related to the Dirichlet form analysis on the Sierpiński gasket. We discuss the analog of the classical Laplacian; approximation by harmonic functions that gives a notion of a gradient; directional energies and an equipartition of energy; analysis with respect to the energy measure; harmonic coordinates; and non self-similar Dirichlet fo...

Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket

We use spectral decimation to provide formulae for computing the harmonic tangents and gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products.

Sampling on the Sierpinski Gasket

Eigenvalues of the Conformal Laplacian

We introduce a differential topological invariant for compact differentiable manifolds by counting the small eigenvalues of the Conformal Laplace operator. This invariant vanishes if and only if the manifold has a metric of positive scalar curvature. We show that the invariant does not increase under surgery of codimension at least three and we give lower and upper bounds in terms of the α-genus.