First eigenvalues of geometric operators under the Ricci flow

Abstract

In this paper, we prove that the first eigenvalues of − Δ + c R -\Delta + cR ( c ≥ 1 4 c\geq \frac 14 ) are nondecreasing under the Ricci flow. We also prove the monotonicity under the normalized Ricci flow for the cases c = 1 / 4 c= 1/4 and r ≤ 0 r\le 0 .