EVOLUTION AND MONOTONICITY FOR A CLASS OF QUANTITIES ALONG THE RICCI-BOURGUIGNON FLOW

Abstract

. In this paper we consider the monotonicity of the lowest constant λ ba ( g ) under the Ricci-Bourguignon flow and the normalized Ricci- Bourguignon flow such that the equation − ∆ u + au log u + bRu = λ ba ( g ) u with (cid:82) M u 2 dV = 1 , has positive solutions, where a and b are two real con-stants. We also construct various monotonic quantities under the Ricci- Bourguignon flow and the normalized Ricci-Bourguignon flow. Moreover, we prove that a compact steady breather which evolves under the Ricci- Bourguignon flow should be Ricci-flat.