$L^p$-spectrum of the Dirac operator on products with hyperbolic spaces

TL;DR

This paper investigates the $L^p$-spectrum of the Dirac operator on complete manifolds, revealing that it can depend on $p$ and providing explicit calculations for hyperbolic spaces and their products.

Contribution

It explicitly computes the $L^p$-spectrum for hyperbolic spaces and their products, demonstrating cases where $p$-independence of the spectrum fails.

Findings

$L^p$-spectrum depends on $p$ for hyperbolic spaces.

Explicit formulas for the $L^p$-spectrum of hyperbolic spaces.

Identification of $p$-dependence in the spectrum on product manifolds.

Abstract

We study the $L^p$-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on $p$. As a first example where $p$-independence fails we compute explicitly the $L^p$-spectrum for the hyperbolic space and its product with compact spaces.