2011/11/21 by Tobias Colding, Tobias Holck Colding, Colding, Tobias Holck · 9 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Bounded function #Curvature #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Metric Geometry (math.MG) #Monotonic function #Pure mathematics #Ricci curvature #Ricci flow #Tangent #Tangent bundle #Tangent cone #Tangent space #Uniqueness #math.AP #math.DG #math.MG
paper · pdf · doi:10.48550/arxiv.1111.4715
published in arXiv (Cornell University) (Cornell University)
arxiv created 2011/11/21 · openalex publication_date 2011/11/21 · arxiv updated 2011/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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. 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 Li-Yau. 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 Schoen-Yau and Witten.