Geometriae Dedicata

, Volume 133, Issue 1, pp 169–179 | Cite as

Curvature integrals under the Ricci flow on surfaces

Original Paper


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)



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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Graduate School of Pure and Applied SciencesUniversity of TsukubaTsukubaJapan

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