The connectedness conjecture for the essential spectrum of the Laplacian

Let $M$ be a complete noncompact Riemannian manifold whose Ricci tensor is bounded below, meaning

$$\operatorname{Ric}(.,.) \geq c\,g(.,.).$$

Connectedness conjecture. The essential spectrum of $-\triangle_g$ on functions is a connected subset of the positive real line $[a,\infty)$.

Under mild geometric conditions, the Laplacian is expected not to have positive eigenvalues of finite multiplicity. This conjecture asserts connectedness of the essential spectrum under a lower Ricci-curvature bound; the source attributes it to earlier work, but no resolution is given here.