The hyperbolic geometric flow on Riemann surfaces

De-Xing Kong Kefeng Liu De-Liang Xu

Geometric Analysis and Geometric Topology mathscidoc:1912.43064

Communications in Partial Differential Equations, 34, (6), 553-580, 2009.5
In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on <sup>2</sup> in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric <i>g</i> <sub> <i>ij</i> </sub> keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably large", then the solution metric <i>g</i> <sub> <i>ij</i> </sub> converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature <i>R</i>(<i>t</i>, <i>x</i>) goes to positive infinity as (<i>t</i>, <i>x</i>) tends to the blowup points, and a flow with surgery has
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@inproceedings{de-xing2009the,
  title={The hyperbolic geometric flow on Riemann surfaces},
  author={De-Xing Kong, Kefeng Liu, and De-Liang Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111334503635624},
  booktitle={Communications in Partial Differential Equations},
  volume={34},
  number={6},
  pages={553-580},
  year={2009},
}
De-Xing Kong, Kefeng Liu, and De-Liang Xu. The hyperbolic geometric flow on Riemann surfaces. 2009. Vol. 34. In Communications in Partial Differential Equations. pp.553-580. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221111334503635624.
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