TY - JOUR
T1 - Isoperimetric rigidity and distributions of 1-Lipschitz functions
AU - Nakajima, Hiroki
AU - Shioya, Takashi
N1 - Funding Information:
The second named author is partially supported by a Grant-in-Aid for Scientific Research Grant Number 26400060 from the Japan Society for the Promotion of Science. The authors would like to thank Prof. Shouhei Honda for his valuable comments.
Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/6/20
Y1 - 2019/6/20
N2 - We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.
AB - We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-Émery Ricci curvature.
KW - Concentration of measure
KW - Isoperimetric profile
KW - Lipschitz function
KW - Metric measure space
KW - Observable variance
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U2 - 10.1016/j.aim.2019.04.043
DO - 10.1016/j.aim.2019.04.043
M3 - Article
AN - SCOPUS:85064947005
VL - 349
SP - 1198
EP - 1233
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -