Eigenvalues of the drifted Laplacian on complete metric measure spaces

In this paper, first we study a complete smooth metric measure space (M-n, g, e(-f) dv) with the (infinity)-Bakry-Emery Ricci curvature Ric(f) >= a/2 g for some positive constant a. It is known that the spectrum of the drifted Laplacian Delta(f) for M is discrete and the first nonzero eigenvalue of Delta(f) has lower bound a/2. We prove that if the lower bound a/2 is achieved with multiplicity k >= 1, then k <= n, M is isometric to Sigma(n-k) x R-k for some complete (n - k)-dimensional manifold S and by passing an isometry, (M-n, g, e(-f) dv) must split off a gradient shrinking Ricci soliton (R-k, g(can), a/4 |t|(2)), t is an element of R-k. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian L = Delta-1/2 < x, del.> for properly immersed self-shrinkers in the Euclidean space Rn+ p, p >= 1 and show the discreteness of the spectrum of L and a logarithmic Sobolev inequality.