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We compare ambient and outer Lipschitz geometry of Lipschitz normally embedded H\"older triangles in $\mathbb{R}^4$. In contrast to the case of $\mathbb{R}^3$ there are infinitely many equivalence classes. The equivalence classes are related to the so-called microknots.

The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fern\'andez de Bobadilla at al. and it captures deep Lipschitz phenomena. However, to become a definitive powerful tool, it must be widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for the inner and outer metrics. We completely determine the MD-Homology of surfaces for the inner metric and we present a great variety of interesting MD-Homology of surfaces for the outer metric, for instance, we determine the MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we explicit the diversity of MD-Homology of surfaces for the outer metric in general, showing how hard it is to completely solve the outer classification problem. On the other hand, we show that, under specific conditions, the weakly outer Lipschitz equivalence determines completely the MD-Homology of surfaces for the outer metric, showing that these two subjects are quite related., Comment: 36 pages

In this note, we present some results on the Mattei's conjecture, which asserts that the algebraic multiplicity of one-dimensional holomorphic foliations is a topological invariant. For instance, we establish a version of the A'Campo-L\^e Theorem adapted for foliations, which was originally formulated for hypersurfaces. More precisely, we show that, under certain conditions, the algebraic multiplicity equals to 1 of holomorphic one-dimensional foliations is a topological invariant. As a consequence, we prove that saddle-nodes are also topological invariant. Additionally, we address a fundamental result in the literature concerning Mattei's conjecture proved by Rosas-Bazan, which says that under the existence of a homeomorphism extending throughout a neighborhood of the exceptional divisor of the first blowing up, Mattei's conjecture has a positive answer. As a consequence of a characterization of the invariance of multiplicity proved here, we obtain that Mattei's conjecture has a positive answer if the homeomorphism extends locally in a small neighborhood where there is a separatrix not contained in the exceptional divisor of the first blowing up, but not necessarily extends throughout a neighborhood of whole exceptional divisor., Comment: 14 pages

For a foliation by CMC hypersurfaces on a complete Riemannian manifold $M^{n+1}$ with sectional curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{K_0}$, under certain additional hypotheses, for instance, if $M$ is a compact (without boundary) or if ${\rm div} (\nabla_N N)=0$ where the gradient of the function ${\rm div} (N)$ is non zero, then we prove that $|H|\equiv \sqrt{K_0}$, where $N$ is a unit vector field orthogonal to the foliation. Moreover, in the the case that $M$ is compact, we prove also that all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where was proved the above result in the case $K_0=0$ and $M$ compact. Under the same additional hypotheses as above, we obtain that for a foliation by CMC hypersurfaces on a complete Riemannian manifold $M$ with Ricci curvature bounded from below by $-nK_0\leq 0$, the mean curvature $H$ of the leaves of the foliation satisfies $|H|\leq \sqrt{K_0}$. This gives a positive partial answer to a conjecture due to Meeks III, P\'erez and Ros. We also provide some partial answers to several other problems., Comment: We have found a mistake in our proof. Our proof only works if we add some hypothesis on the foliation, for instance, if N(H) is constant on the leaves or div(\nabla_N N)=0 where the gradient of H is not zero. Thus, the conjectures are still open. We slightly rewrote the text and changed the title. Subsection 4.1 and Section 5 are new. 31 pages

In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of $\mathbb{C}^2$ as an algebraic surface. For instance, we prove that a complex analytic set that is log-Lipschitz regular at 0 (i.e., a complex analytic set that has a neighbourhood of the origin which bi-log-Lipschitz homeomorphic to an Euclidean ball) must be smooth at 0. We prove even more, we prove that if a complex analytic set $X$ such that, for each $0<\alpha<1$, $(X,0)$ and $(\mathbb{R}^k,0)$ are bi-$\alpha$-H\"older homeomorphic, then $X$ must be smooth at 0. These results generalize the Lipschitz Regularity Theorem, which says that a Lipschitz regular complex analytic set must be smooth. Global versions of these results are also presented here and, in particular, we obtain a characterization of an affine linear subspace as a pure-dimensional entire complex analytic set., Comment: Typos and a mistake in Proposition 3.2 have been corrected. 23 pages

We show that two bi-Lipschitz equivalent Brieskorn-Pham hypersurfaces have the same multiplicities at $0$. Moreover we show that if two algebraic $(n-1)$-dimensional cones $P, R\subset\mathbb C^n$ with isolated singularities are homeomorphic, then they have the same degree., Comment: 5 pages

In this article, we show the H\"older invariance of the Henry-Parusinski invariant. For a single germ $ f$, the Henry-Parusinski invariant of $ f $ is given in terms of the leading coefficients of the asymptotic expansion of $ f $ along the branches of the generic polar curve of $f$. As a consequence, we obtain that the classification problem of polynomial function-germs, with uniformly bounded degree, under H\"older equivalence, admits continuous moduli., Comment: 19 pages

In this paper, we prove the following version of the famous Bernstein's theorem: Let $X\subset \mathbb R^{n+k}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$. Then, the following statements are equivalent: (1) $X$ is an affine linear subspace; (2) $X$ is a definable set that is Lipschitz regular at infinity and its geometric tangent cone at infinity, $C(X,\infty)$, is a linear subspace; (3) $X$ is a definable set, blow-spherical regular at infinity and $C(X,\infty)$ is a linear subspace; (4) $X$ is a definable set that is Lipschitz normally embedded at infinity and $C(X,\infty)$ is a linear subspace; (5) the density of $X$ at infinity is 1. Consequently, we prove the following generalization of Bernstein's theorem: Let $X\subset \mathbb R^{n+1}$ be a closed and connected set with Hausdorff dimension $n$. Assume that $X$ satisfies the monotonicity formula at $p\in X$ and there are compact sets $K\subset \mathbb{R}^n$ and $\tilde K\subset \mathbb{R}^{n+1}$ such that $X\setminus \tilde K$ is a minimal hypersurface that is the graph of a $C^2$-smooth function $u\colon \mathbb{R}^{n}\setminus K\to \mathbb{R}$. Assume that $u$ has bounded derivative whenever $n>7$. Then $X$ is a hyperplane. Several other results are also presented. For example, we generalize the o-minimal Chow's theorem, we prove that any entire complex analytic set that is bi-Lipschitz homeomorphic to a definable set in an o-minimal structure must be an algebraic set. We also obtain that Yau's Bernstein Problem, which says that an oriented stable complete minimal hypersurface in $\mathbb{R}^{n+1}$ with $n\leq 6$ must be a hyperplane, holds true whether the hypersurface is a definable set in an o-minimal structure., Comment: It was added some references and Corollary 3.11. 34 pages and 2 figures

We show that for every $k\ge 3$ there exist complex algebraic cones of dimension $k$ with isolated singularities, which are bi-Lipschitz and semi-algebraically equivalent but they have different degrees. We also prove that homeomorphic projective hypersurfaces with dimension greater than 2 have the same degree. In the final part of the paper, we classify links of real cones with base $\mathbb{P}^1\times \mathbb{P}^2.$ As an application we give an example of three four dimensional real algebraic cones in $\mathbb{R}^8$ with isolated singularity which are semi-algebraically and bi-Lipschitz equivalent but they have non-homeomorphic bases., Comment: 13 pages. arXiv admin note: text overlap with arXiv:2302.05387

In this article, we prove that for a definable set in an o-minimal structure with connected link (at 0 or infinity), the inner distance of the link is equivalent to the inner distance of the set restricted to the link. With this result, we obtain several consequences. We present also several relations between the local and the global Lipschitz geometry of singularities. For instance, we prove that two sets in Euclidean spaces, not necessarily definable 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., Comment: The article was reorganized and Section 3 is new. 23 pages and 2 figures

In this article, we present a complete classification, with normal forms, of the real algebraic curves under blow-spherical homeomorphisms at infinity., Comment: 15 pages and 3 figures. arXiv admin note: text overlap with arXiv:2302.02026

This article is devoted to studying complex algebraic sets under (global) blow-spherical equivalence. The main results of this article are complete classifications of complex algebraic curves. Firstly, we present a complete classification of complex algebraic curves under blow-spherical homeomorphisms at infinity and, then, we present a complete classification of complex algebraic curves under (global) blow-spherical homeomorphisms. We also present several properties of the blow-spherical equivalence. For instance, we prove that the degree of curves is preserved under blow-spherical homeomorphisms at infinity. Another property presented here is a Bernstein-type result which says that a pure dimensional complex algebraic set which is blow-spherical homeomorphic at infinity to a Euclidean space must be an affine linear subspace., Comment: 27 pages and 8 figures. Some typos were fixed and the Section 5 was added

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is shown that $p$ is a singular point of the curve $C$ if and only if this multiplicity is greater than or equal to 2, in this sense, such an integer number can be considered as a measure of how singular can be a point of the curve $C$. In these notes, we address the classical concept of multiplicity of singular points of complex algebraic sets (not necessarily complex curves) and we approach the nature of the multiplicity of singular points as a geometric invariant from the perspective of the Multiplicity Conjecture (Zariski 1971). More precisely, we bring a discussion on the recent results obtained jointly with Lev Birbrair, Javier Fern\'andez de Bobadilla, L\^e Dung Trang and Mikhail Verbitsky on the bi-Lipschitz invariance of the multiplicity., Comment: 36 pages and 1 figure

We classify semi-algebraic surfaces in $\mathbb{R}^n$ with isolated singularities up to bi-Lipschitz homeomorphisms with respect to the inner distance. In particular, we obtain complete classifications for the Nash surfaces and the complex algebraic curves. We also address the minimal surfaces with finite total curvature., Comment: 20 pages and 2 figures

In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a graph of an entire Lipschitz function must be a hyperplane. H. B. Lawson, Jr. and R. Osserman presented examples showing that an analogous result for arbitrary codimension is not true. In this article, we prove a complex non-parametric version of Moser's Bernstein Theorem. More precisely, we prove that any entire complex analytic set in $\mathbb{C}^n$ which is Lipschitz regular at infinity must be an affine linear subspace of $\mathbb{C}^n$. In particular, a complex analytic set which is a graph of an entire Lipschitz function must be affine linear subspace. That result comes as a consequence of the following characterization of algebraic sets, which is also proved here: if $X$ and $Y$ are entire complex analytic sets which are bi-Lipschitz homeomorphic at infinity then $X$ is a complex algebraic set if and only if $Y$ is a complex algebraic set too. Thus, an entire complex analytic set is a complex algebraic set if and only if it is bi-Lipschitz homeomorphic at infinity to a complex algebraic set. No restrictions on the singular set, dimension nor codimension are required in the results proved here., Comment: This paper was already posted on ResearchGate in April 2021

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.

In this paper, we prove Fukui-Kurdyka-Paunescu's Conjecture, which says that subanalytic arc-analytic bi-Lipschitz homeomorphisms preserve the multiplicities of real analytic sets. We also prove several other results on the invariance of the multiplicity (resp. degree) of real and complex analytic (resp. algebraic) sets. For instance, still in the real case, we prove a global version of Fukui-Kurdyka-Paunescu's Conjecture. In the complex case, one of the results that we prove is the following: If $(X,0)\subset (\mathbb{C}^n,0), (Y,0)\subset (\mathbb{C}^m,0)$ are germs of analytic sets and $h\colon (X,0)\to (Y,0)$ is a semi-bi-Lipschitz homeomorphism whose graph is a complex analytic set, then the germs $(X,0)$ and $(Y,0)$ have the same multiplicity. One of the results that we prove in the global case is the following: If $X\subset \mathbb{C}^n, Y\subset \mathbb{C}^m$ are algebraic sets and $\phi\colon X\to Y$ is a semialgebraic semi-bi-Lipschitz homeomorphism such that the closure of its graph in $\mathbb{P}^{n+m}(\mathbb{C})$ is an orientable homological cycle, then ${\rm deg}(X)={\rm deg}(Y)$., Comment: 19 pages. Some typos fixed and a small change in the title

This article is devoted to studying multiplicity and regularity of real analytic sets. We present an equivalence for real analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On regularity, we show that blow-spherical regularity of real analytic implies $C^1$ smoothness only in the case of real analytic curves. On multiplicity, we present a generalization for Gau-Lipman's Theorem about differential invariance of the multiplicity in the complex and real cases, we show that the multiplicity ${\rm mod}\,2$ is invariant by blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. We present also a complete classification of the germs of real analytic curves., Comment: 32 pages, 2 figures

A path-connected subanalytic subset in $\mathbb{R}^n$ is naturally equipped with two metrics: the inner and the outer metrics. We say that 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. It is a fundamental question to know if the LNE property is conical, i.e., if it is possible to describe the LNE property of a germ of a subanalytic set in terms of the properties of its link. We answer this question by introducing a new notion called link Lipschitz normally embedding. We prove that this notion is equivalent to the LNE notion in the case of sets with connected links., Comment: The previous version had two sets of different results. We have divided our pre-print into two, and this version contains the first set of results, which was sections 1-4. The second part, which was Section 5, will appear in a new arXiv submission. The title was changed

This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of Gau-Lipman's Theorem, i.e., it is proved that the multiplicity mod 2 of real analytic sets is a differential invariant. We prove also a generalization of Gau-Lipman's Theorem.

We introduce a new metric homology theory, Moderately Discontinuous Homology, which captures Lipschitz properties of metric subanalytic germs. The main novelty is to allow "moderately discontinuous" chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group $MDH^b_\bullet$ for any $b\in [1,\infty]$ and homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ for any $b\geq b'$. Here $b$ is a "discontinuity rate". The homology groups for the inner or outer metric are proved to be finitely generated and that only finitely many homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ are essential. For $b=1$ it recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty$ the $MD$- homology recovers the homology of the punctured germ. Hence, our invariant interpolates from the germ to its tangent cone. Our homology theory is a bi-Lipschitz subanalitic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer-Vietoris long exact sequences. Moreover, fixed a discontinuity rate $b$ we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are $b$-moderated; this makes the theory quite flexible. In the complex analytic setting we introduce an enhancement called Framed MD Homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD Homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs., Comment: 65 pages

Facing the ever-growing demand for data storage will most probably require a new paradigm. Nanoscale magnetic skyrmions are anticipated to solve this issue as they are arguably the smallest spin textures in magnetic thin films in nature. We designed cobalt-based multilayered thin films where the cobalt layer is sandwiched between two heavy metals providing additive interfacial Dzyaloshinskii-Moriya interactions, which reach a value close to 2 mJ m-2 in the case of the Ir|Co|Pt asymmetric multilayers. Using a magnetization-sensitive scanning x-ray transmission microscopy technique, we imaged small magnetic domains at very low field in these multilayers. The study of their behavior in perpendicular magnetic field allows us to conclude that they are actually magnetic skyrmions stabilized by the large Dzyaloshinskii-Moriya interaction. This discovery of stable sub-100 nm individual skyrmions at room temperature in a technologically relevant material opens the way for device applications in a near future.