Beschreibung

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima’s ergodic theorem for the harmonic functions in the domain of the Lp generator. Secondly we prove analogues of Yau’s and Karp’s Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain growth criteria must be constant. As consequence we give an integral criterion for recurrence.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

The present book deals with the spectral geometry of infinite graphs. This topic involves the interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat flow of the underlying graph. These three subjects are brought together under the unifying perspective of Dirichlet forms. The spectral geometry of manifolds is a well-established field of mathematics. On manifolds, the focus is on how Riemannian geometry, the spectral theory of the Laplace–Beltrami operator, Brownian motion and heat evolution interact. In the last twenty years large parts of this theory have been subsumed within the framework of strongly local Dirichlet forms. Indeed, this point of view has proven extremely fruitful.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schrödinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß-Bonnet theorem.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces

In this article we prove upper bounds for the k-th Laplace eigenvalues below the essential spectrum for strictly negatively curved Cartan–Hadamard manifolds. Our bound is given in terms of k^2 and specific geometric data of the manifold. This applies also to the particular case of non‐compact manifolds whose sectional curvature tends to minus infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.

Related project(s):
59Laplacians, metrics and boundaries of simplicial complexes and Dirichlet spaces