Sharp gradient estimates and monotonicity in positive Ricci curvature
Cosmin Manea
TL;DR
This work extends sharp gradient estimates for Green's functions and related monotonicity formulae from open manifolds with non-negative Ricci curvature to closed manifolds with positive Ricci curvature. Central to the approach is the Schrödinger operator $L=-\Delta + \frac{n(n-2)k}{4}$ and its Green's function, from which a curvature-adapted distance function $b$ is defined via $b=2\operatorname{arcsn}_k(G^{1/(2-n)}/2)$. The authors establish a sharp gradient bound and a suite of unparametrized and one-parameter monotonicity relations for functionals $A$, $V$, $A_\beta$, and $V_\beta$, yielding rigidity characterizations of the model spaces $\mathbb{S}^n_k$ and, notably, a new proof of Bishop's volume comparison in dimension four. Applications include volume and area bounds for level sets of $b$ and a priori control of Green's-function-driven quantities, linking analytic estimates to classical comparison theorems in Riemannian geometry.
Abstract
We prove a sharp gradient estimate for the natural Green's function of a closed manifold with positive Ricci curvature. We also show that this estimate is closely related to a family of monotonicity formulae. These results extend those previously obtained by Colding and Minicozzi for open manifolds with non-negative Ricci curvature. We further obtain several geometric applications, including a new proof of Bishop's volume comparison theorem in dimension four.