Global existence and convergence of solutions of Calabi flow on surfaces of genus \(h\geq 2\) (Q5930010)

Let \((\Sigma, g_ 0)\) be a closed Riemann surface and assume \(g(t)\) is a family of metrics on \(\Sigma\) solving the Calabi flow \(dg/dt=\Delta R\cdot g, g(0)=g_ 0\) on a maximal time interval \([0, T)\). The author presents the result that the Calabi flow admits a solution on \([0,\infty)\) for any arbitrary background metric \(g_ 0\) which converges to a metric of constant curvature \(R\), if the genus of \(\Sigma\) is at least \(2\). The crucial estimate in this paper is the Bondi-mass type estimate NEWLINE\[NEWLINE(d/dt)\int_\Sigma e^{3\lambda}\, d\mu_ 0\leq C_ 1(g_ 0, R_ 0) +C_ 2(g_ 0, R_ 0)\int_\Sigma e^{-\lambda}\, d\mu_ 0,NEWLINE\]NEWLINE where \(\lambda\) is chosen such that \(g(t)=e^{2\lambda}g_ 0\), which is possible because the flow preserves the conformal class. This estimate should be compared with a similar Bondi-mass loss formula obtained by \textit{P. T. Chruściel} [Commun. Math. Phys. 137, no. 2, 289--313 (1991; Zbl 0729.53071)] cited by the author.