Comparison theorems on smooth metric measure spaces with boundary

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Lin Feng Wang ◽  
Ze Yu Zhang ◽  
Yu Jie Zhou
Keyword(s):  
Mean Curvature ◽  
Comparison Theorems ◽  
Metric Measure Spaces ◽  
Weighted Mean ◽  
Dirichlet Eigenvalue ◽  
Weighted Mean Curvature ◽  
First Dirichlet Eigenvalue ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Weighted Laplacian

AbstractIn this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

scholarly journals Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abimbola Abolarinwa ◽  
Akram Ali ◽  
Ali Alkhadi
Keyword(s):  
Lower Bound ◽  
Vector Fields ◽  
First Eigenvalue ◽  
Metric Measure Spaces ◽  
Cheeger Constant ◽  
Weighted Mean Curvature ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Geometric Constant ◽  
Weighted Manifolds

AbstractWe establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

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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