FIRST EIGENVALUES OF GEOMETRIC OPERATORS UNDER THE YAMABE FLOW

Abstract

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator <TEX>$-{\Delta}_{\phi}+{\frac{R}{2}}$</TEX> under the Yamabe flow, where <TEX>${\Delta}_{\phi}$</TEX> is the Witten-Laplacian operator, <TEX>${\phi}{\in}C^2(M)$</TEX>, and R is the scalar curvature with respect to the metric g(t). As a consequence, we construct some monotonic quantities under the Yamabe flow.