Author:
Honda Shouhei,Peng Yuanlin
Abstract
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-parabolic
$\operatorname {RCD}(0,\,N)$
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
$x$
;
\[ |\nabla \mathsf{b}_x|(y) \le \left(N(N-2)\nu_x\right)^{\frac{1}{N-2}}, \quad \text{for any }y \in X \setminus \{x\}. \]
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
$\operatorname {RCD}(N-2,\, N-1)$
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
$\operatorname {RCD}(0,\,N)$
spaces. It is emphasized that the almost rigidities are also proved, which are new even in the smooth framework.
Publisher
Cambridge University Press (CUP)
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