16th Panhellenic Geometry Conference

The Lp-spectrum of the Laplacian on forms over warped products and Kleinian groups.

27 Sept 2024, 17:40

40m

Amphitheatre "Konstantin Karatheodory" (Department of Mathematics, NKUA)

Speaker

Dr Petros Siasos (University of Cyprus)

Description

In this talk we will present some recent results concerning the $L^p$-spectrum of the Laplacian on $k$-forms. We will generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we will see that the $L^p$-spectrum of the Laplacian on $k$-forms contains a parabolic region which depends on $k$, $p$ and the limiting curvature $a_0$ at infinity. For quotients $M={\mathbb{H}}^{N+1}/\mathrm{\Gamma }$ with $\mathrm{\Gamma }$ a geometrically finite group such that $M$ has infinite volume and no cusps, we will show that the $L^p$-spectrum of the Laplacian on $k$-forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line. This talk is based on my PhD thesis at the University of Cyprus.