An overview on calculus and heat flow in metric measure spaces and spaces with Riemannian curvature bounded from below

2013 ◽  
pp. 1-25 ◽  
Author(s):  
Luigi Ambrosio
Keyword(s):  
Heat Flow ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

scholarly journals Regularity of Lagrangian flows over RCD*(K, N) spaces

2020 ◽  
Vol 2020 (765) ◽  
pp. 171-203 ◽  
Author(s):  
Elia Brué ◽  
Daniele Semola
Keyword(s):  
Vector Fields ◽  
Additional Assumption ◽  
Natural Extension ◽  
Regularity Result ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Regularity Results ◽  
Symmetric Derivative

AbstractThe aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolev vector fields over compact metric measure spaces verifying the Riemannian curvature dimension condition. We first prove, borrowing some ideas already present in the literature, that flows generated by vector fields with bounded symmetric derivative are Lipschitz, providing the natural extension of the standard Cauchy–Lipschitz theorem to this setting. Then we prove a Lusin-type regularity result in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular) therefore extending the already known Euclidean result.

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