Original Paper

Geometriae Dedicata

, Volume 133, Issue 1, pp 169-179

First online:

Curvature integrals under the Ricci flow on surfaces

  • Takumi YokotaAffiliated withGraduate School of Pure and Applied Sciences, University of Tsukuba Email author 

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In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.


Ricci flow Total absolute curvature Total curvature Asymptotic volume ratio

Mathematics Subject Classification (2000)