On link of Lipschitz normally embedded sets

Mendes, Rodrigo E.; Sampaio, Jaime

doi:10.48550/arxiv.2101.05572

Cited by 3 publications

(6 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See other characterizations in Subsection 3.1. In particular, we recover the main results proved in [20] and [21]. Some other direct consequences of Theorem 3.1 are the following: Corollary 3.5.…”

Section: Introductionsupporting

confidence: 74%

See 1 more Smart Citation

On Lipschitz Geometry at infinity of complex analytic sets

Sampaio

2023

Calc. Var.

View full text Add to dashboard Cite

In this article, we prove that for a definable set in an ominimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restrict to the link. As consequences, we obtain: ( 1) a definable set, with connected link at infinity, is LNE at infinity if and only if it is LLNE at infinity; (2) a definable set is LNE at infinity if and only if its stereographic modification is LNE at the North Pole; (3) a connected definable set is LNE if and only if its stereographic modification is LNE; and under certain extra conditions we prove that: (4) two definable sets are definably inner (resp. outer) lipeomorphic if and only if their stereographic modifications are definably inner (resp. outer) lipeomorphic if and only if their inversions are definably inner (resp. outer) lipeomorphic. Moreover, we also prove that two sets in Euclidean spaces, not necessarily definables in an o-minimal structure, are outer lipeomorphic if and only if their stereographic modifications are outer lipeomorphic if and only if their inversions are outer lipeomorphic.

show abstract

Section: Introductionsupporting

confidence: 74%

“…Similarly, we also have the local version of the above result which is an easy adaptation of the main result in [20] and was already proved in [21].…”

Section: Bymentioning

confidence: 57%

On Lipschitz Geometry at infinity of complex analytic sets

Sampaio

2023

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…First of all, it is clear that (4) ⇒ (1) − (3). Since X LNE at 0 and π 1 (L 0 (X)) ∼ = 0, by Theorem 4.1 in [22], X is LLNE at 0. Let X 0 = C(X, 0) ∩ S 2m−1 and X t := ( 1 t X) ∩ S 2m−1 t for t > 0.…”

Section: On Characterization Of Smoothnessmentioning

confidence: 94%

On characterization of smoothness of complex analytic sets

Fernandes,

Sampaio

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness.In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of normal surfaces. For example, we prove that a complex analytic set, with an isolated singularity at 0, is smooth at 0 if and only if it is locally metrically conical at 0 and its link at 0 is a homotopy sphere.

show abstract

“…In [MS21], Mendes and Sampaio proved a broad generalization of Proposition 1.20 which provides a characterization of LNE subanalytic germs via their links. This result was further generalized by Nguyen in [Ngu21] to any definable set in a o-minimal structure (not necessarily polynomially bounded).…”

Section: First Examplesmentioning

confidence: 98%

“…Remark 1.24. Condition (2) is stated in [MS21] in the case where the function ρ equals the distance to the origin. In that case, X r = S n−1 r ∩ X is the link of (X, 0) at distance r and a germ (X, 0) satisfying condition (2) is said to be link-LNE (or simply LLNE ).…”

Section: First Examplesmentioning

confidence: 99%

On Lipschitz Normally Embedded singularities

Fantini¹,

Pichon²

2022

Preprint

View full text Add to dashboard Cite

Any subanalytic germ (X, 0) ⊂ (R n , 0) is equipped with two natural metrics: its outer metric, induced by the standard Euclidean metric of the ambient space, and its inner metric, which is defined by measuring the shortest length of paths on the germ (X, 0). The germs for which these two metrics are equivalent up to a bilipschitz homeomorphism, which are called Lipschitz Normally Embedded, have attracted a lot of interest in the last decade. In this survey we discuss many general facts about Lipschitz Normally Embedded singularities, before moving our focus to some recent developments on criteria, examples, and properties of Lipschitz Normally Embedded complex surfaces. We conclude the manuscript with a list of open questions which we believe to be worth of interest.This paper is dedicated to Walter Neumann, wonderful friend and outstanding mathematician.

show abstract

On link of Lipschitz normally embedded sets

Cited by 3 publications

References 12 publications

On Lipschitz Geometry at infinity of complex analytic sets

On Lipschitz Geometry at infinity of complex analytic sets

On characterization of smoothness of complex analytic sets

On Lipschitz Normally Embedded singularities

Contact Info

Product

Resources

About