On the relation between Ricci-Harmonic solitons and Ricci solitons

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• 作者：Meng Zhu^a ; ^b ; ^{mzhu@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ricci-Harmonic flow ; Ricci-Harmonic soliton ; Ricci soliton
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：882-889
• 全文大小：289 K

文摘

Let (M^m,g_ij)$(M^m,g_{ij})$ and (Nⁿ,h_βγ)$(N^n,h_{\beta \gamma })$ be two Riemannian manifolds, and $\varphi :M\to N$ a smooth map. By definition, a gradient Ricci-Harmonic soliton satisfies for some f∈C^∞(M)$f\in C^{\infty }(M)$ and constants α and λ  . Here τ_gϕ=tr_g(∇dϕ)${\tau }_g\varphi =t{r}_g(\mathrm{\nabla }d\varphi )$ is the tension filed of ϕ  . We prove that when α>0$\alpha >0$ and the sectional curvature of N   is bounded from above by $\frac{\alpha }{m}$, any shrinking or steady Ricci-Harmonic soliton (i.e., λ>0$\lambda >0$ or λ=0$\lambda =0$, respectively) must be a Ricci soliton, namely, ϕ is a constant map. In particular, it implies that the shrinking and steady solitons generated from Bernhard List's flow [9] are exactly the corresponding solitons of the Ricci flow, and hence some recent results regarding the shrinking solitons of List's flow are actually duplications of the previous results for Ricci solitons.