arXiv subjects

José Edson Sampaio

Publications and source records attributed to José Edson Sampaio.

At least 19 recordsLinked to original sources

arXiv2026

On the Arnold corank, the Zariski multiplicity, and the tangent Milnor fiber problems

In this article, we approach the Arnold corank problem posed by Arnold himself in 1975, which asks whether the corank of a holomorphic function is a topological invariant. Here, the corank of a holomorphic function is the corank of its Hessian matrix at 0, which is one of the most basic and fundamental invariants of a holomorphic function. Here, we prove the topological invariance of the corank mod 2. Consequently, we obtain another proof of the Arnold corank problem in dimension 3, which was recently proved by Fernandes, Jelonek, and the author of this article. We also prove that the corank is constant in a continuous family of holomorphic functions with isolated singularities and constant Milnor number. We also approach a conjecture recently posed by Budur, Fernández de Bobadilla, Lê, and Nguyen (2022), which states that if two germs of holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms are homotopy equivalent. We prove that a (much) weaker version of this conjecture, which states that if two holomorphic functions are right topologically equivalent, then the Milnor fibers of their initial forms have the same Euler characteristic, is equivalent to the Zariski multiplicity conjecture for right topological equivalence. Some other results are also presented here. For instance, the multiplicity $> 2$ of generalized semi-Bireskorn-Pham hypersurfaces is an invariant for the right topological equivalence.

math.AG↗

arXiv2026

CMC foliations on Euclidean spaces are minimal foliations

In this article, we provide a complete answer to one of the most classical conjectures in differential geometry (of foliations). More precisely, we present a complete positive answer to the classical conjecture, due to Meeks III, Pérez, and Ros (2008), that states that every foliation on $\mathbb{R}^{n+1}$ by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if $n\leq 6$, such a CMC foliation must consist of parallel hyperplanes.

math.DG↗

arXiv2026

Metric-geometric Chow theorem

In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set $X\subset \mathbb{C}^n$ that is definable in an o-minimal structure on $\mathbb{R}$ must be an algebraic set. This result is known nowadays as the o-minimal Chow theorem. In this article, we present some geometric and metric versions of Chow's theorem that generalize the o-minimal Chow's theorem. For instance, we prove that if $X\subset \mathbb{C}^n$ is a pure $d$-dimensional entire complex analytic set, then the following statements are equivalent: (1) $X$ is a complex algebraic set; (2) $\mathcal{H}^{2d+1}(C(X,\infty))=0$, where $\mathcal{H}^{k}(A)$ denotes the $k$-dimensional Hausdorff measure of $A$, and $C(X,\infty)$ denotes the tangent cone at infinity of $X$; (3) $\mathcal{H}^{2d+2}(C_{\mathbb{C}}(X,\infty))=0$, where $C_{\mathbb{C}}(X,\infty)$ denotes the complex tangent cone at infinity of $X$; (4) $\mathcal{H}^{2d}(Z(X,\infty))=0$, where for $A\subset \mathbb{C}^k$, $Z(A,\infty)=\{[v]\in \mathbb{C}P^{k-1};v\in C_{\mathbb{C}}(A,\infty)\}$; (5) For any $k\in\{d+1,...,n\}$ and for any projection $π\colon \mathbb{C}^{n}\to \mathbb{C}^{k}$ such that $π^{-1}(0)\cap C_{\mathbb{C}}(X,\infty)=\{0\}$ and $Y=π(X)$, $\mathcal{H}^{2d}(Z(Y,\infty))<\mathcal{H}^{2d}(\mathbb{C}P^{d})$.

math.AG↗

arXiv2026

Bi-Lipschitz equivalent cones with different degrees

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.

math.AG↗

arXiv2026

On smoothness, tangent cones, and the metric geometry of definable sets

In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations presented by Ghomi and Howard (2014) and by Kurdyka, Le Gal, and Nhan (2018). For instance, we prove that for any $X\subset \mathbb{R}^n$ that is a locally closed $d$-dimensional definable set in an o-minimal structure, the following items are equivalent: (1) $X$ is Lipschitz normally embedded (LNE), $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$ and depends continuously on $p$; (2) For each $p\in X$, $X$ is Lipschitz regular at $p$ and $C_4(X,p)$ is a $d$-dimensional linear subspace; (3) $X$ is a topological manifold and for each $p\in X$, $X$ is LNE at $p$ and $C_4(X,p)=C_3(X,p)$; (4) $X$ is a topological manifold, and $C_5(X,p)$ is a $d-$dimensional subset for any $p\in X$; (5) $X$ is $C^1$ smooth.

math.MG↗

arXiv2026

Real and bi-Lipschitz versions of the Theorem of Nobile

The renowned Theorem of Nobile, proved by Nobile in 1975, states that a pure dimensional complex analytic set $X$ is analytically smooth if and only if its Nash transformation $η: \mathcal{N}(X) \to X$ is an analytic isomorphism. While the Theorem of Nobile was fundamental in complex geometry, it remained an open question for 50 years whether the theorem held for real analytic sets, even more so for cases that demand only $C^{k}$ smoothness. This paper presents a proof for the real version of the Theorem of Nobile, even under $C^{k}$ smoothness conditions. Specifically, we prove that for a pure dimensional real analytic set $X$ the following statements are equivalent: (1) $X$ is a real analytic (resp. $C^{k+1,1}$) submanifold; (2) the mapping $η\colon\mathcal{N}(X)\to X$ is a real analytic (resp. $C^{k,1}$) diffeomorphism; (3) the mapping $η\colon\mathcal{N}(X)\to X$ is a $C^{\infty}$ (resp. $C^{k,1}$) diffeomorphism; (4) $X$ is a $C^{\infty}$ (resp. $C^{k+1,1}$) submanifold. Consequently, we prove the bi-Lipschitz version of the Theorem of Nobile, demonstrating that a complex analytic set $X$ is analytically smooth if and only if its Nash transformation $η\colon \mathcal{N}(X)\to X$ is a homeomorphism that is locally bi-Lipschitz. A sharp version of this theorem, which holds in the much more general setting of locally definable sets in an o-minimal structure, is also presented here.

math.AG↗

arXiv2025

Proof of the metric Arnold's corank problem

In this article, we approach the Arnold corank problem, posed by Arnold in 1975, which asks whether the corank of holomorphic functions is an ambient topological invariant. Here, we obtain a complete positive answer to the metric Arnold corank problem, which asks whether the corank of holomorphic functions is an ambient bi-Lipschitz invariant. Consequently, we show that for complex hypersurfaces, the multiplicity equal to two is an ambient bi-Lipschitz invariant. We also prove that the Arnold corank problem holds true for holomorphic functions of three variables. Other topological invariants are also presented.

math.CV↗

arXiv2025

Lipschitz Geometry of Mixed Polynomials

We investigate the (ambient) bi-Lipschitz V-equivalence of two-variable mixed polynomials satisfying the Newton inner non-degeneracy condition. Concerning triviality, we show that ambient bi-Lipschitz V-triviality for families $\{f + \varepsilon θ\}_{\varepsilon \in \mathbb{R}}$ is guaranteed when $f$ is semi-radially weighted homogeneous and the weighted radial degree of every monomial in $θ$ is greater than the weighted radial degree associated with $f$. However, in the general case, we prove that it is not guaranteed, even though ambient topological V-triviality still holds. For the classification problem, we define two simple metric links and prove that they suffice to determine bi-Lipschitz V-equivalence within the class of mixed polynomials that are $Γ_{\rm inn}$-nice. A key outcome is that neither the Newton boundary $Γ(f)$ nor the C-face diagram $Γ_{\rm inn}$ constitutes an invariant of this equivalence for such mixed polynomials. To outcome this, we introduce new data extracted from the two face diagrams under consideration and prove that, under certain generic conditions, these data become fundamental invariants for the bi-Lipschitz equivalences. This provides a fundamental step toward a bi-Lipschitz classification of these mixed polynomials.

math.MG↗

arXiv2025

Bounds to the mean curvature of leaves of CMC foliations

The main goal of this present paper is to bring the results proved by Barbosa, Kenmotsu and Oshikiri (1991) and its ideas to a perspective where the Ricci curvature is bounded from below. For instance, for a foliation by CMC hypersurfaces on a compact (without boundary) Riemannian manifold $M^{n+1}$ with Ricci curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{K_0}$, we prove that $|H|\equiv \sqrt{K_0}$ and all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where the above result was proved in the case $K_0=0$. We also obtain a proof of the following: for a foliation by CMC hypersurfaces on a compact (without boundary) 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 $|H|\leq \sqrt{K_0}$. Furthermore, if the foliation contains a leaf $L$ whose absolute mean curvature is $|H_L|=\sqrt{K_0}$, then either $K_0=0$ and all the leaves of $\mathfrak{F}$ are totally geodesic, or $K_0>0$ and there is a totally umbilical leaf.

math.DG↗

arXiv2025

Lipschitz geometry of the image of finite mappings

This paper is devoted to the study of the LNE property in complex analytic hypersurface parametrized germs, that is, the sets that are images of finite analytic map germs from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$. We prove that if the multiplicity of $f$ is equal to his generic degree, then the image of $f$ is LNE at 0 if and only if it is a smooth germ. We also show that every finite corank 1 map is sattisfies the previous hypothesis. Moreover, we show that for an injective map germ $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{n+1},0)$, the image of $f$ is LNE at 0 if and only if $f$ is an embedding.

math.AG↗

arXiv2025

On the Invariance of the Real Milnor Number under Asymptotically Lipschitz Equivalence

We investigate sufficient conditions for the invariance of the real Milnor number under $\mathcal{R}$-bi-Lipschitz equivalence for function-germs $ f, g \colon (\mathbb{R}^n, 0) \to (\mathbb{R}, 0) $. More generally, we explore its invariance within the extended framework of $\mathcal{R}$-asymptotically Lipschitz equivalence. To this end, we introduce the $α$-derivative, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of $C^k$ and $C^{\infty}$ equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.

math.AG↗

arXiv2025

On the Milnor fibres of initial forms of topologically equivalent holomorphic functions

Budur, Fernandes de Bobadilla, Le and Nguyen (2022) conjectured that if two germs of holomorphic functions are topologically equivalent, then the Milnor fibres of their initial forms are homotopy equivalent. In this note, we give affirmative answers to this conjecture in the case of plane curves. We show also that a positive answer to this conjecture implies in a positive answer to the famous Zariski multiplicity conjecture both in the case of right equivalence or in the case of hypersurfaces with isolated singularities.

math.AG↗

arXiv2024

On the topological invariance of the algebraic multiplicity of holomorphic foliations

In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of A'Campo-Lê's Theorem for foliations, i.e., the algebraic multiplicity equal to one is a topological invariant in dimension two. This result is further generalized to higher dimensions under mild conditions; as a consequence, we prove that saddle-nodes are topologically invariant. We prove that the algebraic multiplicity is a topological invariant in several classes of foliations that contain, for instance, the generalized curves and the foliations of second type. Additionally, we address a fundamental result by Rosas-Bazan, which states that the existence of a homeomorphism extending through a neighborhood of the exceptional divisor of the first blow-up implies the topological invariance of the algebraic multiplicity. We show that the result holds if the homeomorphism extends locally near a singularity, even if it does not extend over the entire divisor.

math.CV↗

arXiv2024

Log-Lipschitz and Hölder regularity imply smoothness for complex analytic sets

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 $0<α<1$, $(X,0)$ and $(\mathbb{R}^k,0)$ are bi-$α$-Hölder 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.

math.AG↗

arXiv2024

Universality theorem for LNE Hölder triangles

We compare ambient and outer Lipschitz geometry of Lipschitz normally embedded Hölder 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.

math.AG↗

arXiv2024

Moderately discontinuous homology of real surfaces

The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fernández 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.

math.MG↗

arXiv2024

On metric equivalence of the Brieskorn-Pham hypersurfaces

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.

math.AG↗

arXiv2024

On the Moser's Bernstein Theorem

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.

math.DG↗