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New monotonicity formulas for Ricci curvature and applications. I

Tobias Holck Colding Department of Mathematics, Massachusetts Institute of Technology

Differential Geometry mathscidoc:1701.10004

Acta Mathematica, 209, (2), 229-263, 2011.11
We prove three new monotonicity formulas for manifolds with a lower Ricci curvature bound and show that they are connected to rate of convergence to tangent cones. In fact, we show that the derivative of each of these three monotone quantities is bounded from below in terms of the Gromov–Hausdorff distance to the nearest cone. The monotonicity formulas are related to the classical Bishop–Gromov volume comparison theorem and Perelman’s celebrated monotonicity formula for the Ricci flow. We will explain the connection between all of these.
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@inproceedings{tobias2011new,
  title={New monotonicity formulas for Ricci curvature and applications. I},
  author={Tobias Holck Colding},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203359717120762},
  booktitle={Acta Mathematica},
  volume={209},
  number={2},
  pages={229-263},
  year={2011},
}
Tobias Holck Colding. New monotonicity formulas for Ricci curvature and applications. I. 2011. Vol. 209. In Acta Mathematica. pp.229-263. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203359717120762.
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