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

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.