scholarly journals Quantitative Regularity for p-Minimizing Maps Through a Reifenberg Theorem

2021 ◽  
Author(s):  
Mattia Vedovato
Keyword(s):  
Riemannian Manifolds ◽  
Upper Bound ◽  
Type Theorem ◽  
Regularity Result ◽  
Singular Points ◽  
Singular Set ◽  
Minkowski Content ◽  
Minimizing Maps ◽  
Quantitative Regularity ◽  
Appl Math

AbstractIn this article we extend to arbitrary p-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $$p=2$$ p = 2 . We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done in Cheeger and Naber (Commun Pure Appl Math 66(6): 965–990, 2013). Then, adapting the work of Naber and Valtorta (Ann Math (2) 185(1): 131–227, 2017), we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is k-rectifiable for a k which only depends on p and the dimension of the domain.

Holonomy of Sub-Riemannian Manifolds

1997 ◽  
Vol 08 (03) ◽  
pp. 317-344 ◽  
Author(s):  
Elisha Falbel ◽  
Claudio Gorodski ◽  
Michel Rumin
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Smooth Manifold ◽  
Symmetric Spaces ◽  
Type Theorem ◽  
Central Symmetry ◽  
Contact Type ◽  
Riemannian Symmetric Spaces

A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).

scholarly journals Image sets of folding surfaces

2005 ◽  
Vol 2005 (1) ◽  
pp. 91-97
Author(s):  
Ana M. d'Azevedo Breda
Keyword(s):  
Riemannian Manifolds ◽  
Special Class ◽  
Singular Set ◽  
Combinatorial Properties ◽  
Image Set

Isometric foldings are a special class of length-preserving maps of Riemannian manifolds and were initially studied by S. Robertson. For an explanation of their topological and combinatorial properties, see the related works of Ana Breda, Altino Santos, M. El-Ghoul, and E. M. Elkholy. Here, we explore some properties of the singular set and describe the image set of planar, spherical, and hyperbolic foldings.

scholarly journals A Decomposition Theorem for Immersions of Product Manifolds

2015 ◽  
Vol 59 (1) ◽  
pp. 247-269 ◽  
Author(s):  
Ruy Tojeiro
Keyword(s):  
Riemannian Manifolds ◽  
Space Form ◽  
Type Theorem ◽  
Decomposition Theorem ◽  
Product Manifold ◽  
Isometric Immersions ◽  
Product Metrics ◽  
Special Cases ◽  
De Rham ◽  
Shape Operators

AbstractWe introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally or globally decomposed as a product manifold endowed with a polar metric. For such a product manifold, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapted to its product structure in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as of its extension by Nölker to isometric immersions of warped products.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

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