First Eigenvalues of Geometric Operators under the Ricci Flow

TL;DR

This paper proves that the first eigenvalues of a specific geometric operator are nondecreasing under Ricci flow, with particular cases showing monotonicity under normalized flow, advancing understanding of spectral geometry evolution.

Contribution

It establishes the monotonicity of the first eigenvalues of the operator $- riangle + cR$ under Ricci flow for certain constants, extending spectral analysis in geometric flows.

Findings

First eigenvalues of $- riangle + cR$ are nondecreasing under Ricci flow for $c \\geq 1/4$.

Monotonicity also holds under normalized flow for $c=1/4$ and $r \\leq 0$.

Results contribute to spectral geometry and geometric analysis of Ricci flow.

Abstract

In this paper, we prove that the first eigenvalues of $-\Delta + cR$ ($c\geq \frac14$) is nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized flow for the case $c=1/4$, and $r\le 0$.