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The $L^p$-spectrum of the Laplacian on forms over warped products and Kleinian groups

Petros Siasos

Abstract

In this article, we 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 prove 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 $M=\mathbb{H}^{N+1}/Γ$ with $Γ$ a geometrically finite group such that $M$ has infinite volume and no cusps, we prove 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.

The $L^p$-spectrum of the Laplacian on forms over warped products and Kleinian groups

Abstract

In this article, we generalize the set of manifolds over which the -spectrum of the Laplacian on -forms depends on . 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 prove that the -spectrum of the Laplacian on -forms contains a parabolic region which depends on , and the limiting curvature at infinity. For with a geometrically finite group such that has infinite volume and no cusps, we prove that the -spectrum of the Laplacian on -forms is a exactly a parabolic region together with a set of isolated eigenvalues on the real line.
Paper Structure (7 sections, 32 theorems, 184 equations)

This paper contains 7 sections, 32 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. The $L^p$ spectrum for a class of warped product metrics part I
  3. The $L^p$ spectrum for a class of warped product metrics part II
  4. The $L^p$ spectrum on a class of Kleinian Groups
  5. Appendix
  6. Curvatures in warped products
  7. Heat kernel on forms

Key Result

Theorem 1.1

Let $M$ be a warped product at infinity where the warping function $f \in B$. For any $0\leq k\leq \frac{n}{2}$ and $1\leq p \leq 2$, the $L^p$ spectrum of the Hodge Laplacian, $\sigma (p,k, \Delta)$ contains $Q_{p,k}$. The remaining cases for $p$ and $k$ are given by duality as $\sigma (p,k, \Delta

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Definition 3.1
  • Theorem 3.1
  • Proposition 3.1
  • proof
  • ...and 38 more