Hausdorff Convergence of Riemannian Manifolds and Its Applications

2018 ◽  
Author(s):  
Kenji Fukaya
Keyword(s):  
Riemannian Manifolds ◽  
Hausdorff Convergence

scholarly journals Essential circles and Gromov–Hausdorff convergence of covers

2016 ◽  
Vol 08 (01) ◽  
pp. 89-115
Author(s):  
Conrad Plaut ◽  
Jay Wilkins
Keyword(s):  
Riemannian Manifolds ◽  
Minimal Cardinality ◽  
Geodesic Space ◽  
Limiting Behavior ◽  
Hausdorff Convergence ◽  
Generating Sets ◽  
Finite Set ◽  
Geodesic Spaces ◽  
Simply Connected ◽  
Gromov Hausdorff Convergence

The [Formula: see text]-covers of Sormani–Wei ([20]) are known not to be “closed” with respect to Gromov–Hausdorff convergence. In this paper we use the essential circles introduced in [19] to define a larger class of covering maps of compact geodesic spaces called “circle covers” that are “closed” with respect to Gromov–Hausdorff convergence and include [Formula: see text]-covers. In fact, we use circle covers to completely understand the limiting behavior of [Formula: see text]-covers. The proofs use the descrete homotopy methods developed by Berestovskii, Plaut, and Wilkins, and in fact we show that when [Formula: see text], the Sormani–Wei [Formula: see text]-cover is isometric to the Berestovskii–Plaut–Wilkins [Formula: see text]-cover. Of possible independent interest, our arguments involve showing that “almost isometries” between compact geodesic spaces result in explicitly controlled quasi-isometries between their [Formula: see text]-covers. Finally, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.

On Ricci Riemannian Manifolds

2010 ◽  
Vol 0 (-1) ◽  
pp. 437-446 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Riemannian Manifolds

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

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals Local Regularity for the Harmonic Map and Yang–Mills Heat Flows

2021 ◽  
Author(s):  
Ahmad Afuni
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Map ◽  
Local Regularity ◽  
Yang Mills ◽  
Heat Flows ◽  
Singular Sets ◽  
Regularity Results ◽  
Local Monotonicity ◽  
Partial Differ ◽  
Singular Time

AbstractWe establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows at the first singular time. Finally, we show that smooth blow-ups at rapidly forming singularities of these flows are necessarily nontrivial and admit a positive lower bound on their heat ball energies. These results crucially depend on some local monotonicity formulæ for these flows recently established by Ecker (Calc Var Partial Differ Equ 23(1):67–81, 2005) and the Afuni (Calc Var 555(1):1–14, 2016; Adv Calc Var 12(2):135–156, 2019).

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