Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2016; 53(4): 1113-1122

Printed July 31, 2016

https://doi.org/10.4134/BKMS.b150530

Copyright © The Korean Mathematical Society.

First eigenvalues of geometric operators under the Yamabe flow

Shouwen Fang and Fei Yang

Yangzhou University, China University of Geosciences

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

MSC numbers: 53C21, 53C44