The maximum principle and the Yamabe flow (Q2759788)

Let \(M\) be a complete non-compact manifold of dimension \(n\geq 3\). The authors deal with the maximum principle and the Yamabe flow on \(M\). Assume that \(M\) has Ricci curvature bounded from below and has uniform bound on its scalar curvature. Then, it is proved that there exists a global solution \(u\) to Yamabe flow on \(M\). Moreover, for all \(0\leq T<+\infty\), all the derivatives of \(u\) are uniformly bounded on \(M\times [0,T]\). If \(M\) is locally conformal flat with Ricci curvature bounded from below, then there exists \(T_0>0\) such that the Yamabe flow has a solution \(u\) on \(M\times [0,T_0]\). All the derivatives of \(u\) are uniformly bounded on \(M\times [0,T_0]\). NEWLINENEWLINENEWLINEThe proof goes through a maximum principle on non-compact manifolds. A Harnack estimate for the Yamabe flow is also obtained. Methods due to \textit{B. Chow} and \textit{R. S. Hamilton} are used [Invent. Math. 129, No. 2, 213-238 (1997; Zbl 0903.58054)].NEWLINENEWLINENEWLINERecall that the Yamabe problem on compact manifolds was solved by \textit{N. S. Trudinger} [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 22, 265-274 (1968; Zbl 0159.23801)], \textit{T. Aubin} [`Nonlinear Analysis on Manifolds', Springer-Verlag, Berlin (1982; Zbl 0512.53044)] and \textit{R. Schoen} [J. Differ. Geom. 20, 479-495 (1984; Zbl 0576.53028)].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00056].