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

Published online by Cambridge University Press:  17 January 2024

Shouhei Honda  and
Yuanlin Peng
Shouhei Honda
Affiliation:
Mathematical Institute, Tohoku University, Japan ([email protected])
Yuanlin Peng
Affiliation:
Mathematical Institute, Tohoku University; JSPS Research Fellow, Japan ([email protected])
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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.

Keywords

Ricci curvaturemetric measure spaceGreen functionsrigidityGromov–Hausdorff convergenceOptimal transportation theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 54
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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