Metrically Un-knotted Corank 1 Singularities of Surfaces in 
                
                  
                
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Birbrair, Lev; Mendes, Rodrigo; Nuño–Ballesteros, J. J.

doi:10.1007/s12220-017-9973-2

Cited by 8 publications

(5 citation statements)

References 9 publications

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“…The notion of LNE first appeared in a paper of Birbrair and Mostowski [5], it has since been an active research area and many interesting results were proved, see, for example, [1–5, 9–13, 19–21]. Recently, Mendes and Sampaio [18] gave a nice criterion for the germ of a subanalytic set to be LNE based on the LNE condition on the link.…”

Section: Introductionmentioning

confidence: 99%

On a link criterion for Lipschitz normal embeddings among definable sets

Nhan

2023

Mathematische Nachrichten

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It is known by a result of Mendes and Sampaio that the Lipschitz normal embedding of a subanalytic germ is fully characterized by the Lipschitz normal embedding of its link. In this note, we show that the result still holds for definable germs in any o-minimal structure on (ℝ, +, .). We give an example showing that for homomorphisms between MD-homologies induced by the identity map, being isomorphic is not enough to ensure that the given germ is Lipschitz normally embedded. This is a negative answer to the question asked by Bobadilla et al. in their paper about moderately discontinuous homology. K E Y W O R D S definable sets, Lipschitz normal embeddings, MD-homology, o-minimal structures M S C ( 2 0 2 0 ) 14B05, 32C05 INTRODUCTIONGiven a connected definable set 𝑋 ⊂ ℝ 𝑛 , one can equip 𝑋 with two natural metrics: the outer metric (𝑑 𝑜𝑢𝑡 ) induced by the Euclidean metric of the ambient space ℝ 𝑛 and the inner metric (𝑑 𝑖𝑛𝑛 ), where the distance between two points in 𝑋 is defined as the infimum of the lengths of rectifiable curves in 𝑋 connecting these points. We call 𝑋 Lipschitz normally embedded (or LNE for brevity) if these two metrics are equivalent, that is, there is a constant 𝐶 > 0 such that ∀𝑥, 𝑦 ∈ 𝑋, 𝑑 inn (𝑥, 𝑦) ≤ 𝐶𝑑 out (𝑥, 𝑦).Any such 𝐶 is referred to as a LNE constant for 𝑋. We say that 𝑋 is LNE at a point 𝑥 0 ∈ 𝑋 (or the germ (𝑋, 𝑥 0 ) is LNE) if there is a neighborhood 𝑈 of 𝑥 0 in ℝ 𝑛 such that 𝑋 ∩ 𝑈 is LNE. It is worth noticing that, in the o-minimal setting, the definitions of connected and path connected set are equivalent, and every continuous definable curve connecting two points is rectifiable, that is, it has finite length. The notion of LNE first appeared in a paper of Birbrair and Mostowski [5], it has since been an active research area and many interesting results were proved, see, for example, [1-5, 9-13, 19-21]. Recently, Mendes and Sampaio [18] gave a nice criterion for the germ of a subanalytic set to be LNE based on the LNE condition on the link. Namely, they prove that 2958

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Section: Introductionmentioning

confidence: 99%

On a link criterion for Lipschitz normal embeddings among definable sets

Nhan

2023

Mathematische Nachrichten

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“…This definition was introduced by L. Birbrair and T. Mostowski [5], where they just call it normally embedded. As it was already remarked in [18], Lipschitz Normal Embedding is a very active research area with many recent results giving necessary conditions for a set to be LNE in the real and complex setting, e.g., by L. Birbrair, M. Denkowski, A. Fernandes, D. Kerner, F. Misef, W. D. Neumann, J. J. Nuño-Ballesteros, H. M. Perdersen, A. Pichon, M. A. S. Ruas, M. Tibar etc ( [4], [6], [9], [14], [17], [18] and [19]). Recent works show that LNE property appears in several fields of Mathematics, e.g.,…”

Section: Introductionmentioning

confidence: 99%

“…• Topology: (Knot Theory) It was proved in [4] that, for a large class of locally LNE analytic parametrized surfaces, the knots presented as the link of such surfaces is always trivial (unknotted); (Fundamental group) It was proved in [10] that if two compact subanalytic sets with same LNE constant and which are close enough with respect to Hausdorff distance, then their fundamental groups are isomorphic.…”

Section: Introductionmentioning

confidence: 99%

On link of Lipschitz normally embedded sets

Mendes¹,

Sampaio²

2021

Preprint

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A path connected subanalytic subset in R n is naturally equipped with two metrics, the inner and the outer metrics. We say that such a subset is Lipschitz normally embedded (LNE) if these two metrics are equivalent. In this article we give some criteria for a subanalytic set to be LNE. We introduce a new notion called link Lipschitz normally embedded and we prove that this notion is equivalent to LNE notion in the case of sets with connected links and we present some applications of it and, in particular, we prove that the LNE property is a conical property. Some examples are also presented.

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“…Let us now prove (2). For a moment, we just assume that (X, 0) is a superisolated singularity and we prove a lemma.…”

Section: 4mentioning

confidence: 99%

“…Lipschitz Normal Embedding (LNE) is a very active research area with many recent results, e.g., by Birbrair, Fernandes, Kerner, Mendes, Neumann, Nuño-Ballesteros, Pedersen, Pichon, Ruas, Sampaio ( [2], [6], [8], [10], [14]), including a characterization of LNE for semialgebraic sets ( [1]) and a characterization of LNE for complex surfaces ( [13]).…”

Section: Introductionmentioning

confidence: 99%

Lipschitz normal embedding among superisolated singularities

Misev,

Pichon

2018

Preprint

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Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities which is radically different from the class of minimal singularities.

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Metrically Un-knotted Corank 1 Singularities of Surfaces in $$\mathbb {R}^4$$ R 4

Cited by 8 publications

References 9 publications

On a link criterion for Lipschitz normal embeddings among definable sets

On a link criterion for Lipschitz normal embeddings among definable sets

On link of Lipschitz normally embedded sets

Lipschitz normal embedding among superisolated singularities

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