Analysis on fibred cusp spaces

TL;DR

This survey explores the geometric and analytic properties of fibred cusp spaces, a broad class of non-compact manifolds, focusing on spectral geometry, boundary problems, and microlocal analysis techniques.

Contribution

It provides a comprehensive overview of results on fibred cusp spaces, highlighting the differences between incomplete and complete cases, and sketches proof ideas for key theorems.

Findings

Analysis of spectral properties of fibred cusp spaces

Results on analytic torsion and index theory in these spaces

Microlocal analysis of resolvent and heat kernel

Abstract

We give a survey of analytic and geometric results on `fibred cusp spaces', a large class of non-compact Riemannian manifolds which include the regular parts of singular spaces with incomplete cusp singularities as well as complete spaces with asymptotically hyperbolic cusp or asymptotically Euclidean structures at infinity. These results cover topics in spectral geometry, in particular analytic torsion and index theory, and boundary value problems. The underlying tools include a careful microlocal analysis of the resolvent and the heat kernel. We include an exposition of the geometric and analytic foundations and sketch the ideas of the proofs of the main theorems. Special emphasis is put on the common features of and the differences between the incomplete and various kinds of complete settings.