On the essential spectrum of the Laplacian and the drifted Laplacian

This paper concerns the L2L2 essential spectrum of the Laplacian Δ and the drifted Laplacian ΔfΔf on complete Riemannian manifolds endowed with a weighted measure e−fdvolge−fdvolg. We prove that the essential spectrum of the drifted Laplacian ΔfΔf is [0,∞)[0,∞) provided the Bakry–Émery curvature tensor RicfRicf is nonnegative and f   has sublinear growth. When Ricf⩾12g and |∇f|2⩽f|∇f|2⩽f, we show that the essential spectrum of the Laplacian is also [0,∞)[0,∞). During the proofs of these results, the f-volume growth estimate plays an important role and may be of independent interest.