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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1116–1135 | Cite as

Monotonicity of Eigenvalues and Functionals Along the Ricci–Bourguignon Flow

  • Lin Feng WangEmail author
Article
  • 155 Downloads

Abstract

In this paper we first prove the monotonicity of the lowest eigenvalue of the Schrödinger operator
$$\begin{aligned} \frac{(1-(n-1)\rho )^2}{1-2(n-1)\rho }R-4\varDelta \end{aligned}$$
along the Ricci–Bourguignon flow
$$\begin{aligned} \frac{\partial \mathrm {g}}{\partial t}=-2(\mathrm {Ric}-\rho R\mathrm {g}) \end{aligned}$$
based on an evolving formula of the \(\mathcal {F}_{\rho }\)-functional, and rule out nontrivial steady breathers. Then we prove the monotonicity of the lowest eigenvalue of the Schrödinger operator \(BR-4\varDelta \) along the Ricci–Bourguignon flow for any constant B satisfying
$$\begin{aligned} B\ge \frac{4(1-(n-1)\rho )^2-n\rho }{4(1-2(n-1)\rho )}>0 \end{aligned}$$
for the case that \(\rho \le 0.\) We also study the evolving formula of the \(\mathcal {W}_{\rho }\)-functional and get the monotonicity of the infimum of the \(\mathcal {W}_{\rho }\)-functional, based on which we can prove that a shrinking breather should be an Einstein metric for the case that \(\rho <0.\)

Keywords

Ricci–Bourguignon flow Eigenvalue \(\mathcal {F}_{\rho }\)-functional \(\mathcal {W}_{\rho }\)-functional Evolving formula Breather 

Mathematics Subject Classification

53C21 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.School of ScienceNantong UniversityNantongChina

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