Curvature integrals under the Ricci flow on surfaces Original Paper

First Online: 05 March 2008 Received: 04 December 2007 Accepted: 13 February 2008 DOI :
10.1007/s10711-008-9241-5

Cite this article as: Yokota, T. Geom Dedicata (2008) 133: 169. doi:10.1007/s10711-008-9241-5 Abstract 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.

Keywords Ricci flow Total absolute curvature Total curvature Asymptotic volume ratio

References 1.

Chow B. (1991). The Ricci flow on the 2-sphere.

J. Differ. Geom. 33(2): 325–334

MATH 2.

Cheeger J. and Colding T. (1997). On the structure of spaces with Ricci curvature bounded below.

I. J. Differ. Geom. 46(3): 406–480

MATH MathSciNet 3.

Chow B. and Knopf D. (2004). The Ricci Flow: An Introduction. AMS, Providence, RI

MATH 4.

Chu S.C. (2007). Type II ancient solutions to the Ricci flow on surfaces.

Commun. Anal. Geom. 15(1): 195–215

MATH 5.

Dai X. and Ma L. (2007). Mass under the Ricci flow.

Commun. Math. Phys. 274(1): 65–80

MATH CrossRef MathSciNet 6.

Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. With appendices by Katz, M., Pansu, P., Semmes, S. Progress in Mathematics, 152. Birkhuser Boston, Inc., Boston, MA (1999)

7.

Hamilton, R.: The Ricci flow on Surfaces, Mathematics and General Relativity (Santa Cruz, CA, 1986), pp. 237–262. Contemp. Math., 71, AMS, Providence, RI (1988)

8.

Hamilton, R.: The Formation of Singularities in the Ricci flow, Surveys in Differential Geometry, vol. II (Cambridge, MA, 1993), pp. 7–136, Int. Press, Cambridge, MA (1995)

9.

Huber A. (1957). On subharmonic functions and differential geometry in the large, Comment.

Math. Helv. 32: 13–72

MATH CrossRef MathSciNet 10.

Ji, L., Sesum, N.: Uniformization of conformally finite Riemann surfaces by the Ricci flow. arXiv:math/0703357, preprint

11.

Shi W.X. (1989). Deforming the metric on complete Riemannian manifolds.

J. Differential Geom. 30(1): 223–301

MATH MathSciNet 12.

Shiohama, K., Shioya, T., Tanaka, M.: The Geometry of Total Curvature on Complete Open Surfaces. Cambridge Tracts in Mathematics, 159. Cambridge University Press, Cambridge (2003)

13.

Wu L.F. (1993). The Ricci flow on complete R

^{2} .

Commun. Anal. Geom. 1(3–4): 439–472

MATH 14.

Gromoll D. and Meyer W. (1969). On complete open manifolds of positive curvature.

Ann. Math. 90(2): 75–90

MathSciNet CrossRef © Springer Science+Business Media B.V. 2008

Authors and Affiliations 1. Graduate School of Pure and Applied Sciences University of Tsukuba Tsukuba Japan