Communications in Analysis and Geometry

Volume 28 (2020)

Number 3

Some integral curvature estimates for the Ricci flow in four dimensions

Pages: 707 – 727



Miles Simon (Otto von Guericke University, Magdeburg, Germany)


We consider solutions $(M^4 , g(t)), 0 \leq t \lt T$, to Ricci flow on compact, four-dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and $T \lt \infty$, we show that these estimates imply that the (spatial) integral of the square of the norm of the Riemannian curvature is bounded by a constant independent of time $t$ for all $0 \leq t \lt T$ and that the space time integral over $M \times [0, T)$ of the fourth power of the norm of the Ricci curvature is bounded.

The author gratefully acknowledges the support of SFB TR71 and SPP 2026 of the DFG (German Research Foundation) and the University of Freiburg, where a part of this work was carried out.

Received 14 June 2016

Accepted 8 March 2018

Published 6 July 2020