Sharp gradient estimate, rigidity and almost rigidity of Green functions on non-parabolic $\mathrm{RCD}(0,N)$ spaces

TL;DR

This paper establishes sharp gradient bounds and rigidity results for Green functions on non-parabolic RCD(0,N) spaces, extending classical results and introducing new almost rigidity theorems in a non-smooth setting.

Contribution

It provides the first sharp gradient estimates and rigidity characterizations for Green functions on RCD spaces, extending Colding's results to non-smooth metric measure spaces.

Findings

Sharp gradient upper bounds for Green functions are proven.

Rigidity results characterize when bounds are attained, identifying space isomorphisms.

Almost rigidity results are established, even in smooth cases.

Abstract

Inspired by a result of Colding, the present paper studies the Green function $G$ on a non-parabolic $\mathrm{RCD}(0,N)$ space $(X, \mathsf{d}, \mathfrak{m})$ for some finite $N>2$. Defining $\mathsf{b}_x=G(x, \cdot)^{\frac{1}{2-N}}$ for a point $x \in X$, which plays a role of a smoothed distance function from $x$, we prove that the gradient $|\nabla \mathsf{b}_x|$ has the canonical pointwise representative with the sharp upper bound in terms of the $N$-volume density $\nu_x=\lim_{r\to 0^+}\frac{\mathfrak{m} (B_r(x))}{r^N}$ of $\mathfrak{m}$ at $x$; \begin{equation*} |\nabla \mathsf{b}_x|(y) \le \left(N(N-2)\nu_x\right)^{\frac{1}{N-2}}, \quad \text{for any $y \in X \setminus \{x\}$}. \end{equation*} Moreover the rigidity is obtained, namely, the upper bound is attained at a point $y \in X \setminus \{x\}$ if and only if the space is isomorphic to the $N$-metric measure cone over an $\mathrm{RCD}(N-2, N-1)$ space. In the case when $x$ is an $N$-regular point, the rigidity states an isomorphism to the $N$-dimensional Euclidean space $\mathbb{R}^N$, thus, this extends the result of Colding to $\mathrm{RCD}(0,N)$ spaces. It is emphasized that the almost rigidities are also proved, which are new even in the smooth framework.