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.