scholarly journals Rigidity properties of smooth metric measure spaces via the weighted $p$-Laplacian

2016 ◽  
Vol 145 (3) ◽  
pp. 1287-1299 ◽  
Author(s):  
Nguyen Thac Dung
Keyword(s):  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

A Liouville theorem for weightedp−Laplace operator on smooth metric measure spaces

2016 ◽  
Vol 40 (4) ◽  
pp. 992-1002 ◽  
Author(s):  
Lin Feng Wang ◽  
Ze Yu Zhang ◽  
Liang Zhao ◽  
Yu Jie Zhou
Keyword(s):  
Laplace Operator ◽  
Liouville Theorem ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

scholarly journals Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

2018 ◽  
Vol 6 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Riemannian Manifolds ◽  
Short Note ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Necessary And Sufficient Condition ◽  
Local Dimension ◽  
Base Points ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient

Abstract 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.

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