On the essential spectrum of complete non-compact manifolds

TL;DR

This paper characterizes the essential spectrum of the Laplacian on functions for certain non-compact manifolds, showing it is [0,+∞) under specific curvature conditions, and extends the method to Ricci solitons.

Contribution

It establishes the essential spectrum as [0,+∞) for manifolds with non-negative Ricci curvature at infinity and applies the approach to gradient shrinking Ricci solitons.

Findings

Essential spectrum of Laplacian is [0,+∞) for manifolds with non-negative Ricci curvature at infinity.

Method applies to gradient shrinking Ricci solitons, indicating similar spectral properties.

Provides a spectral analysis framework for non-compact manifolds with specific curvature conditions.

Abstract

In this paper, we prove that the $L^p$ essential spectra of the Laplacian on functions are $[0,+\infty)$ on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative Ricci curvature in many ways.