New monotonicity formulas for Ricci curvature and applications; I
Tobias Holck Colding,
2011.11.21
http://arxiv.org/abs/1111.4715
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 GromovHausdorff
distance to the nearest cone. The monotonicity formulas are related to the
classical BishopGromov volume comparison theorem and Perelman's celebrated
monotonicity formula for the Ricci flow. We will explain the connection between
all of these. Moreover, we show that these new monotonicity formulas are linked
to a new sharp gradient estimate for the Green's function that we prove. This
is parallel to that Perelman's monotonicity is closely related to the sharp
gradient estimate for the heat kernel of LiYau. In [CM4] we will use the
monotonicity formulas we prove here to show uniqueness of certain tangent cones
of Einstein manifolds and in [CM3] we will prove a number of related
monotonicity formulas. Finally, there are obvious parallels between our
monotonicity and the positive mass theorem of SchoenYau and Witten.
