Bottom of the spectrum of complete noncompact Kähler manifolds

Abstract

We present a survey on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds, with particular emphasis on Kähler hyperbolic manifolds and bounded symmetric domains. We also discuss theorems regarding the upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions, along with rigidity results for manifolds attaining the maximal bottom of the spectrum. Throughout the article, we propose several open problems.