New Yamabe-type flow in a compact Riemannian manifold

2021/02/04 by Ma, Li
#35K55 #53C21 #58E35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2102.02398

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+2)/(n-2) and c_n=(4(n-1))/(n-2). We study the Yamabe-type flow u=u(t) satisfying u_t=u^1-p(c_nΔu-ψ(x)u)+r(t)u, in M× (0,T), T>0 with r(t)=∫_M(c_n|∇ u|²+ψ(x)u²)dv/ ∫_Mu^p+1, which preserves the L^p+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 equation c_nΔu-ψ(x)u+r(∞)u^p=0, on M and our result may be considered as a generalization of the result of T.Aubin, Proposition in p.131 in \citeA82.

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