First eigenvalues of geometric operators under the Yamabe flow

Abstract

Suppose $$(M,g_0)$$ is a compact Riemannian manifold without boundary of dimension $$n\ge 3$$ . Using the Yamabe flow, we obtain estimate for the first nonzero eigenvalue of the Laplacian of $$g_0$$ with negative scalar curvature in terms of the Yamabe metric in its conformal class. On the other hand, we prove that the first eigenvalue of some geometric operators on a compact Riemannian manifold is nondecreasing along the unnormalized Yamabe flow under suitable curvature assumption. Similar results are obtained for manifolds with boundary and for CR manifold.