Evolution of a geometric constant along the Ricci flow

Abstract

Abstract In this paper, we establish the first variation formula of the lowest constant λ a b ( g ) $\lambda_{a}^{b}(g)$ along the Ricci flow and the normalized Ricci flow, such that to the following nonlinear equation there exist positive solutions: − Δ u + a u log u + b R u = λ a b u $$-\Delta u+au\log u+bRu=\lambda_{a}^{b} u $$ with ∫ M u 2 d v = 1 $\int_{M}u^{2}\,dv=1$ , where a is a real constant. In particular, the results proved in this paper generalize partial results in Cao (Proc. Am. Math. Soc. 136:4075-4078, 2008) and Li (Math. Ann. 338:927-946, 2007).

Keywords

• Ricci flow
• normalized Ricci flow
• conjugate heat equation