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

Inspired by a result in T. H. Colding. (16).Acta. Math.209(2) (2012), 229-263[16] of Colding, the present paper studies the Green function G on a non-parabolicRCD(0,N)space(X,d,m) for some finite N>2. Defining bx=G(x,<middle dot>)12-Nfor apointx is an element of X, which plays a role of a smoothed distance function from x,we prove that the gradient|del bx|has the canonical pointwise representative with the sharp upper bound in terms of the N-volume density nu x= limr -> 0+m(Br(x))rN of m at x; |del bx|(y)<(N(N-2)nu x)1N-2,for any y is an element of X\{x} Moreover the rigidity is obtained, namely, the upper bound is attained at a pointy is an element of X\{x}if and only if the space is isomorphic to the N-metric measure cone over an RCD(N-2,N-1) space. In the case when xis an N-regular point, the rigidity states an isomorphism to the N-dimensional Euclidean space RN, thus, this extends the result of Colding to RCD(0,N) spaces. It is emphasized that the almost rigidities are also proved, which are new even in the smooth framework.