New Yamabe-type flow in a compact Riemannian manifold

DOI10.1016/J.BULSCI.2023.103244arXiv2102.02398MaRDI QIDQ6359783

Li Ma

Publication date: 3 February 2021

Abstract: In this paper, we set up a new Yamabe type flow on a compact Riemannian manifold $(M,g)$ of dimension $ngeq3$. Let $psi(x)$ be any smooth function on $M$. Let $p=fracn+2n-2$ and $c_n=frac4(n-1)n-2$. We study the Yamabe-type flow $u=u(t)$ satisfying {u_t}=u^{1-p}(c_nDelta u-psi(x)u)+r(t)u, in M imes (0,T), T>0 with r(t)=int_M(c_n| abla u|^2+psi(x)u^2)dv/ int_Mu^{p+1}, which preserves the $L^{p+1}(M)$-norm and we can show that for any initial metric $u_0>0$, the flow exists globally. We also show that in some cases, the global solution converges to a smooth solution to the equation c_nDelta u-psi(x)u+r(infty)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 cite{A82}.

Mathematics Subject Classification ID

Variational inequalities (global problems) in infinite-dimensional spaces (58E35) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Heat and other parabolic equation methods for PDEs on manifolds (58J35) Quasilinear parabolic equations (35K59)

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