Geometriae Dedicata

, Volume 133, Issue 1, pp 169–179

# Curvature integrals under the Ricci flow on surfaces

Original Paper

## 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

53C44

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