Bottom Spectrum Estimate Under Curvature Integrability Condition

TL;DR

This paper establishes an upper bound for the lowest eigenvalue of the Laplacian on manifolds with Ricci curvature controlled in an integral sense, using properties of Green's functions.

Contribution

It introduces a novel method to estimate the bottom spectrum under integral Ricci curvature bounds leveraging Green's function analysis.

Findings

Upper bound for Laplacian spectrum established

Method applicable to manifolds with integral Ricci bounds

Green's function properties are key to the proof

Abstract

In this paper we prove an upper bound for the bottom of the spectrum of the Laplacian on manifolds with Ricci curvature bounded in integral sense. Our arguments rely on the existence of a minimal positive Green's function and its properties.