TY - JOUR

T1 - Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

AU - Honda, Shouhei

N1 - Funding Information:
Acknowledgements: A part of the work is done during the author’s stay in Yau Mathematical Sciences Center (YMSC) at Tsinghua University. The author would like to express his appreciation to Guoyi Xu for his warm hospitality. He is also grateful to YMSC for giving him nice environment. He thanks Luigi Ambrosio, Nicola Gigli, Bangxian Han and Aaron Naber for helpful comments. Moreover, He thanks the referee for the careful reading of the manuscript and for the suggestions in the revision. Finally, he acknowledges the supports of the Grantin-Aid for Young Scientists (B), 16K17585, and of the Grant-in-Aid for Scientific Research (B), 18H01118.

PY - 2018/2/1

Y1 - 2018/2/1

N2 - In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

AB - In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

KW - 53C21

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U2 - 10.1515/agms-2018-0007

DO - 10.1515/agms-2018-0007

M3 - Article

AN - SCOPUS:85056906548

VL - 6

SP - 129

EP - 145

JO - Analysis and Geometry in Metric Spaces

JF - Analysis and Geometry in Metric Spaces

SN - 2299-3274

IS - 1

ER -