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EIGENVALUES OF GEOMETRIC OPERATORS RELATED TO THE WITTEN LAPLACIAN UNDER THE RICCI FLOW

Part of: Global differential geometry

Published online by Cambridge University Press:  27 February 2017

SHOUWEN FANG ,
FEI YANG  and
PENG ZHU
SHOUWEN FANG
Affiliation:
School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225002, P. R. China e-mail: shwfang@163.com
FEI YANG
Affiliation:
School of Mathematics and physics, China University of Geosciences, Wuhan 430074, P. R. China e-mail: yangfei810712@163.com
PENG ZHU
Affiliation:
School of Mathematics and physics, Jiangsu University of Technology, Changzhou, Jiangsu 213001, P. R. China e-mail: zhupeng2004@126.com
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Abstract

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Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C 2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.

MSC classification

Primary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions
Secondary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 743 - 751
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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