The classical de Rham-Hodge theory implies that each cohomology
class of a compact manifold is uniquely represented by a harmonic
form, signifying the important role of Laplacian in geometry. The talk aims
to explain some results relating curvature to the spectrum of Laplacian. We
plan to start by a brief overview for the case of bounded Euclidean domains
and compact manifolds, highlighting some of the fundamental contributions by
Peter Li and others. We then shift our focus to the case of complete
Riemannian manifolds. In particular, it includes our recent joint work with
Ovidiu Munteanu concerning the bottom spectrum of 3-manifolds with scalar
curvature bounded below.