# FIRST EIGENVALUES OF GEOMETRIC OPERATORS UNDER THE YAMABE FLOW

• Fang, Shouwen ;
• Yang, Fei
• Published : 2016.07.31
• 55 9

#### 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 $-{\Delta}_{\phi}+{\frac{R}{2}}$ under the Yamabe flow, where ${\Delta}_{\phi}$ is the Witten-Laplacian operator, ${\phi}{\in}C^2(M)$, 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.

#### Keywords

eigenvalue;Witten-Laplacian;Yamabe flow

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