New Yamabe-type flow in a compact Riemannian manifold

Abstract

In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold (M,g) of dimension n≥3. Let ψ(x) be any smooth function on M. Let p=n+2n−2 and cn=4(n−1)n−2. We study the Yamabe-type flow u=u(t) satisfyingut=u1−p(cnΔu−ψ(x)u)+r(t)u,inM×(0,T),T>0 withr(t)=∫M(cn|∇u|2+ψ(x)u2)dv/∫Mup+1, which preserves the Lp+1(M)-norm and we can show that for any initial metric u0>0, the flow exists globally. We also show that in some cases, the global solution converges to a smooth solution to the equationcnΔu−ψ(x)u+r(∞)up=0,onM and our result may be considered as a generalization of the result of T. Aubin, Proposition in p.131 in [1].