The Dirichlet-Neumann operator for fibred cusp geometries

We consider Riemannian manifolds with boundary where the boundary exhibits singularities of fibred cusp type, or are conformal to these. A simple example is the complement of two touching balls in ${\mathbb{R}}^n$. This type of singularity (at the touching point), in case $n=2$, is often called an incomplete cusp (or horn). Other examples, conformal to these types of spaces and with 'singularity' at infinity, are fundamental domains of Fuchsian groups and uniformly fattened infinite cones in ${\mathbb{R}}^n$.

The Dirichlet-Neumann (DN) operator on a Riemannian manifold with boundary maps Dirichlet boundary data of harmonic functions to their Neumann data. This operator is well studied in the smooth compact case, for example it is known that it is a pseudodifferential operator (PsiDO), and its spectrum (the Steklov eigenvalues) has been studied intensively, as well as the inverse problem for it.

We show that the DN operator for fibred cusp singularities are in a PsiDO calculus adapted to the geometry, the so-called phi-calculus. This yields a precise description of their integral kernels near the singularities. In the talk I will introduce the necessary background on the phi-calculus, and also discuss some of the spectral properties of the DN operator in this setting.

This is joint work with K. Fritzsch und E. Schrohe.