Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems

In this paper we consider piecewise linear (pl) isometric embeddings of Euclidean polyhedra into Euclidean space. A Euclidean polyhedron is just a metric space [Formula: see text] which admits a triangulation [Formula: see text] such that each n-dimensional simplex of [Formula: see text] is affinely isometric to a simplex in 𝔼n. We prove that any 1-Lipschitz map from an n-dimensional Euclidean polyhedron [Formula: see text] into 𝔼3n is ϵ-close to a pl isometric embedding for any ϵ > 0. If we remove the condition that the map be pl, then any 1-Lipschitz map into 𝔼2n + 1 can be approximated by a (continuous) isometric embedding. These results are extended to isometric embedding theorems of spherical and hyperbolic polyhedra into Euclidean space by the use of the Nash–Kuiper C1 isometric embedding theorem ([9] and [13]).

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Sobolev Spaces

10.1093/oso/9780198812050.003.0005 ◽

2018 ◽

Author(s):

D. E. Edmunds ◽

W. D. Evans

Keyword(s):

Sobolev Spaces ◽

Embedding Theorems ◽

Approximation Numbers ◽

Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

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Embedding theorems for Janelidze's matrix conditions

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2019.05.017 ◽

2020 ◽

Vol 224 (2) ◽

pp. 469-506 ◽

Cited By ~ 2

Author(s):

Pierre-Alain Jacqmin

Keyword(s):

Embedding Theorems

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An Intrinsic Characterization of Five Points in a CAT(0) Space

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0111 ◽

2020 ◽

Vol 8 (1) ◽

pp. 114-165

Author(s):

Tetsu Toyoda

Keyword(s):

Metric Space ◽

Isometric Embedding ◽

Metric Spaces ◽

Necessary Conditions ◽

New Approach ◽

Intrinsic Characterization ◽

New Family

AbstractGromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

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Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

Complex Analysis and Operator Theory ◽

10.1007/s11785-021-01089-4 ◽

2021 ◽

Vol 15 (3) ◽

Author(s):

Changbao Pang ◽

Antti Perälä ◽

Maofa Wang

Keyword(s):

Borel Measure ◽

Bergman Spaces ◽

Embedding Theorem ◽

Weighted Bergman Spaces ◽

Embedding Theorems ◽

Positive Borel Measure ◽

Doubling Property ◽

Area Operator ◽

Special Cases ◽

Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

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