Evolution for First Eigenvalue of <i>L</i>_T,f on an Evolving Riemannian Manifold

Abstract

In this paper, evolution formulas for the first non-zero eigenvalue of the operator LT,f on a weighted closed Riemannian manifold along the Ricci flow as well as along the Yamabe flow are formulated. Some monotonic quantities are also derived for the normalized Ricci flow on Bianchi classes.

Keywords

• eigenvalue
• Ricci flow
• Yamabe flow
• homogeneous 3-manifold