Global classical solutions to hyperbolic geometric flow on Riemann surfaces
The authors study the evolution of a Riemann metric \(g_{ij}\) on a Riemann surface \(\mathcal M\) by its Ricci curvature tensor \(R_{ij}\) under the hyperbolic geometric flow equation \[ \frac{\partial^2g_{ij}}{\partial t^2}+2R_{ij}=0. \] The authors are interested in the initial metric \[ ds^2 = u_0(x)(dx^2 + dy^2)\quad \text{at}\quad t = 0 \] on a surface of topological type \(\mathbb R^2\), where \(u_0(x)\) is a \(C^2\) function with bounded \(C^2\) norm and satisfies \[ 0 < k\leq u_0(x)\leq M <\infty. \] The initial data \(u_0(x)\) depends only on \(x\) and not on \(y\). Therefore, the authors consider the Cauchy problem \[ \left\{ \begin{aligned} u_{tt} &-(\ln u)_{xx}=0,\\ u &=u_0(x)\quad\text{at}\quad t=0,\\ u_t &=u_1(x)\quad\text{at}\quad t=0,\end{aligned} \right. \] where \(u_1(x)\in C^1\) with bounded \(C^1\) norm. The authors obtain a necessary and sufficient condition for the global existence of classical solutions to this kind of flows on Riemann surfaces. The results show that any metric on a simply connected non-compact surface converges to one of flat curvature by choosing a suitable initial velocity tensor \(\partial g_{ij}(x, 0)/\partial t\), but the general case of a metric in \(\mathbb R^2\) is still open. If we do not choose a suitable initial velocity tensor \(\partial g_{ij}(x, 0)/\partial t\), the solution to the Cauchy problem will blow up in finite time.
scientific article; zbMATH DE number 5720145