$L^p$-spectral theory for the Laplacian on forms

TL;DR

This work develops a robust $L^p$-spectral theory for the Laplacian on differential $k$-forms on complete noncompact manifolds. By combining heat-kernel bounds, finite propagation-speed arguments for the wave equation, and resolvent analysis, it establishes an $L^p$-Weyl criterion under curvature and volume-regularity hypotheses, and shows the $L^p$-resolvent set lies outside a parabolic region determined by the exponential volume growth. The paper then provides a precise description of the $L^p$-spectrum on hyperbolic space, proving that $\sigma(p,k)$ equals a parabolic region $Q_{p,k}$ with eigenvalue behavior for $p>2$ and no interior eigenvalues for $p eq 2$, along with harmonic-form phenomena in odd dimensions. Overall, the results extend resolvent estimates beyond bounded-geometry assumptions and yield detailed spectral portraits for symmetric spaces, with implications for nonlinear PDEs and geometric analysis on noncompact manifolds.

Abstract

In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and also prove the decomposition of the $L^p$-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on $L^p$ lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space.

Theorems & Definitions (66)

• Theorem 1.1
• Conjecture 1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Lemma 2.2
• Theorem 2.1