scholarly journals Smooth metric measure spaces, quasi-Einstein metrics, and tractors

2012 ◽  
Vol 10 (5) ◽  
pp. 1733-1762 ◽  
Author(s):  
Jeffrey S. Case
Keyword(s):  
Einstein Metrics ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

scholarly journals SMOOTH METRIC MEASURE SPACES AND QUASI-EINSTEIN METRICS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250110 ◽  
Author(s):  
JEFFREY S. CASE
Keyword(s):  
Einstein Metrics ◽  
Conformal Geometry ◽  
Point Of View ◽  
Metric Measure Spaces ◽  
Unified Framework ◽  
Variational Characterization ◽  
Gradient Ricci Solitons ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

Smooth metric measure spaces have been studied from the two different perspectives of Bakry–Émery and Chang–Gursky–Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.

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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