scholarly journals Compactness Theorems for Sequences of Riemannian Manifolds

2019 ◽  
Vol 66 (04) ◽  
pp. 1
Author(s):  
Christina Sormani
Keyword(s):  
Riemannian Manifolds ◽  
Compactness Theorems

scholarly journals Convergence of manifolds and metric spaces with boundary

2018 ◽  
Vol 12 (03) ◽  
pp. 735-774 ◽  
Author(s):  
Raquel Perales
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Level Sets ◽  
Metric Spaces ◽  
Upper Bounds ◽  
Manifolds With Boundary ◽  
Integral Current ◽  
Intrinsic Flat ◽  
Compactness Theorems

We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov–Hausdorff (GH) and Sormani–Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably [Formula: see text] rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require non-negative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.

On the equicontinuity of families of inverse mappings of Riemannian manifolds

2019 ◽  
Vol 16 (4) ◽  
pp. 557-566
Author(s):  
Denis Ilyutko ◽  
Evgenii Sevost'yanov
Keyword(s):  
Riemannian Manifolds ◽  
Type Inequality ◽  
The Given ◽  
Range Of Values

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

Ball-homogeneous and disk-homogeneous Riemannian manifolds

1982 ◽  
Vol 180 (4) ◽  
pp. 429-444 ◽  
Author(s):  
Old?ich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Homogeneous Riemannian Manifolds
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