The Harnack estimate for the modified Ricci flow on complete \(\mathbb{R}^ 2\). (Q1414968)

The authors prove a trace Harnack inequality and a matrix Harnack inequality for the ''modified Ricci flow'' \[ {\partial g\over \partial t} = - {R \over 1+R } \] on \(({\mathbb R}^{2} ,e^{u(t)} g_{\mathbb{R}^{2}})\), where \(g_{\mathbb{R}^{2}}\) is the standard Euclidean metric on \({\mathbb R}^{2} \), and \(R\) is the scalar curvature. This flow was first proposed by S. T. Yau to study the uniformization theorem on complete noncompact surfaces. The Harnack quantity is found by considering expanding gradient solitons for the flow, following \textit{R. S. Hamilton} [J. Diff Geom. 37, 225-243 (1993; Zbl 0804.53023)]. The inequalities are then proved using methods of \textit{R. Hamilton} and \textit{B. Chow} [Commun. Pure Appl. Math. 45, No. 8, 1003--1014 (1992; Zbl 0785.53027)].