Estimates and Monotonicity of the First Eigenvalues Under the Ricci Flow on Closed Surfaces

Abstract

In the paper we first derive the evolution equation for eigenvalues of geometric operator $-\Delta _{\phi }+cR$ under the Ricci flow and the normalized Ricci flow on a closed Riemannian manifold M, where $\Delta _{\phi }$ is the Witten–Laplacian operator, $\phi \in C^{\infty }(M)$, and R is the scalar curvature. We then prove that the first eigenvalue of the geometric operator is nondecreasing along the Ricci flow on closed surfaces with certain curvature conditions when $0<c\le \frac{1}{2}$. As an application, we obtain some monotonicity formulae and estimates for the first eigenvalue on closed surfaces.

Keywords

First eigenvalue / Witten–Laplacian / Ricci flow

Funding

NSFC(11371310, 11401514)

the University Science Research Project of Jiangsu Province(13KJB110029)

National Natural Science Foundation of China(11471145)

the Fundamental Research Funds for the Central Universities (NS2014076)

the Natural Science Foundation of Jiangsu Province(BK20140804)

Qing Lan Project