Quantitative Rigidity Using Colding's Monotonicity Formulas for Ricci Curvature

2025/06/10 by Breiner, Christine, Park, Jiewon
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.08317

Abstract

In \citeColding, Colding proved monotonicity formulas for the Green function on manifolds with nonnegative Ricci curvature. Inspired by the sharp estimates relating the pinching of monotone quantities to the splitting function in \citecjn, in this paper we investigate quantitative control obtained from pinching of Colding's monotone functionals. From the Green functions with poles at (k+1)-many independent points, k-splitting functions are constructed with regularity quantitatively controlled by the pinching. Moreover, the pinching at these independent points controls the distance to the nearest cone of the form ℝ^k × C(X).

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Related

• New monotonicity formulas for Ricci curvature and applications; I
• Sharp gradient estimates and monotonicity in positive Ricci curvature
• Monotonicity formulas for harmonic functions in \rm RCD(0,N) spaces
• Monotonicity formulas and Hessian of the Green function
• Rigidity of the first Betti number via Ricci flow smoothing