The purpose of the paper under review is to study the existence of global Yamabe flows on asymptotically flat manifolds.
Given an \(n\)-dimensional complete manifold \((M^n,g_0)\), \(n\geq 3\), the Yamabe flow on \((M^n,g_0)\) is a family of Riemannian metrics \(\{g(\cdot,t)\}\) on \(M\) defined by the evolution equation \[\begin{cases} \frac{\partial g}{\partial t} &=-Rg\quad \mbox{in}\quad M^n \times [0,T),\\ g(\cdot,0) &= g_0 \quad \mbox{in}\quad M^n, \end{cases}\] where \(R\) is the scalar curvature of the metric \(g:=g(\cdot,t)=u^{\frac{4}{n-2}}g_0\) and \(u: M^n \to \mathbb{R}^+\) is a positive smooth function on \(M^n\).
A Riemannian manifold \(M^n, n\geq 3\) with \(C^{\infty}\) metric \(g\) is called asymptotically flat of order \(\tau > 0\) if there exists a decomposition \(M^n =M_0 \cup M_{\infty}\) with \(M_0\) compact and a diffeomorphism \(M_{\infty} \cong \mathbb{R}^n -B(o,R_0)\) for some constant \(R_0 > 0\) such that \[g_{ij} -\delta_{ij} \in C^{2+\alpha}_{-\tau}(M)\] in the coordinates \(\{x^i\}\) induced on \(M_{\infty}\) and the coordinates \(\{x^i\}\) are called asymptotic coordinates.
First, the author studies the local existence of the Yamabe flow on a complete Riemannian manifold with bounded scalar curvature. Secondly, the author proves the existence of global Yamabe flows on asymptotically flat manifolds.