摘要
In the paper we first derive theevolution equation for eigenvalues of geomet-ric operator-ΔФ+cR under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M,where,is the Witten-Laplacian operator,Ф∈C^(∞)(M),and R is the scalar curvature.We then prove that the first eigenvalue of the geometricoperator is nondecreasing along the Ricci flow on closed surfaces with certain curva-ture conditions when 0<c≤1/2.As an application,we obtain some monotonicityformulae and estimates for the first eigenvalue on closed surfaces.
基金
PRC Grant NSFC(11371310,11401514,11471145)
the University Science Research Project of Jiangsu Province(13KJB110029)
the NaturalScience Foundation of Jiangsu Province(BK20140804)
the Fundamental Research Funds for the CentralUniversities(NS2014076)
Qing Lan Project.
