Spectrum of the Laplacian on manifolds with Spin(9) holonomy

TL;DR

This paper investigates the spectral properties of noncompact manifolds with Spin(9) holonomy, establishing one end and splitting theorems based on the spectrum's bottom, and characterizing harmonic functions with finite Dirichlet integral.

Contribution

It introduces new results on the structure of manifolds with Spin(9) holonomy, including harmonic function characterization and splitting theorems under spectral conditions.

Findings

Harmonic functions with finite Dirichlet integral are Cayley-harmonic.

Established one end result for manifolds with Spin(9) holonomy.

Proved a splitting theorem using Busemann functions.

Abstract

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result. In the second part, we established a splitting type theorem by utilizing the Busemann function..