Searcharxiv⌕ Search

arXiv subjects

Marina Ville

Publications and source records attributed to Marina Ville.

17 recordsLinked to original sources
arXiv2025

Sequences of surfaces in $4$-manifolds

Let $(Σ_n)$ be a sequence of surfaces immersed in a $4$-manifold $M$ which converges to a branched surface $Σ_0$ .\\ We denote by $k^T_p$ (resp. $k^N_p$) the amount of curvature of the tangent bundles $TΣ_n$ (resp. normal bundles $NΣ_n$) which concentrates around a branch point $p$ of $Σ_0$ when $n$ goes to infinity. Alternatively $k^T\pm k^N$ measures how much the twistor degrees drop when we go from $Σ_n$ to $Σ_0$. For complex algebraic curves, $k^T+k^N=0$..\\ In some instances - 1) if $Σ_0$ is made up of at most $3$ branched disks or 2) if $Σ_0$ is area minimizing or 3) if the $Σ_n$'s are minimal - we show that $-k^T\geq |k^N|$ and we investigate the equality case.\\ When the second fundamental forms of the $Σ_n$'s have a common $L^2$ bound, we relate $k^T$ and $k^N$ to the bubbling-off of a current $C$ in the Grassmannian $G_2^+(M)$. If the $Σ_n$'s are minimal, $C$ is a complex curve.

math.DG↗
arXiv2025

Random walks and the symplectic representation of the braid group

We consider the symplectic representation $ρ_n$ of a braid group $B(n)$ in $Sp(2l,\mathbb{Z})$, for $l=\Big[\dfrac{n-1}{2}\Big]$. If $P$ is a polynomial on the $4l^2$ coefficients of the matrices in $Sp(2l,\mathbb{Z})$, we show that the set $\{β\in B(n): P(ρ_n(β))=0\}$ is transient for non degenerate random walks on $B(n)$. We derive that the $n$-braids $β$ which close into a loop $\hatβ$ with $0<|det({\hatβ})|\leq C$ for some constant $C$ form a transient set. And given a prime number $p$, we show that the probability for a given braid to close in a $p$-colorable loop is greater than $\dfrac{1}{p}$. We also derive that for a random $3$-braid, the quasipositive links $(βσ_iβ^{-1}σ_j)^p$ have zero signature for every integer $p$ and $1\leq i,j\leq 2$. \\ As an example of such braids, we investigate the signature of the Lissajous toric knots $3$-braids.

math.GT↗
arXiv2025

Minimal tori in $\mathbb{R}^4$

We describe tools for the study of minimal surfaces in $\mathbb{R}^4$; some are classical (the Gauss maps) and some are newer (the link/braid/writhe at infinity). Then we look for complete proper non holomorphic minimal tori with total curvature $-8π$ and a single end immersed in $\mathbb{R}^4$. We translate the problem into a system of $10$ quadratic or linear equations in $11$ real variables with coefficients in terms of the Weierstrass function $\wp$ and give explicit solutions for these equations if $T$ is a rectangular torus. For the square torus, we have a complete answer with a unique family of solutions generalizing the Chen-Gackstetter torus in $\mathbb{R}^3$. On the other hand, we show that there is no solution on the equianharmonic torus.

math.DG↗
arXiv2025

Biharmonic Hypersurfaces in Euclidean Spaces

An isometric immersion $X: Σ^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic if $Δ^2 X = 0$, i.e. if $ΔH =0$, where $Δ$ and $H$ are the metric Laplacian and the mean curvature vector field of $Σ^n$ respectively. More generally, biconservative hypersurfaces (BCH) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. We study and construct BCH that are holonomic, i.e. for which the principal curvature directions define an integrable net, and we deduce that $Σ^n$ is a holonomic biharmonic hypersurface iff it is minimal.

math.DG↗
arXiv2021

A link at infinity for minimal surfaces in $\mathbb{R}^4$

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the link at infinity can sometimes help us determine if a surface has self-intersection and we illustrate this idea by looking at genus zero surfaces of small total curvature.

math.DG↗
arXiv2021

A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition

We say that a PDE in a Riemannian manifold $M$ is geometric if,$\ $whenever $u$ is a solution of the PDE on a domain $Ω$ of $M$, the composition $u_ϕ:=u\circϕ$ is also solution on $ϕ^{-1}\left( Ω\right) $, for any isometry $ϕ$ of $M.$ We prove that if $u\in C^{1}\left( \mathbb{H}^{n}\right) $ is a solution of a geometric PDE satisfying the comparison principle, where $\mathbb{H}^{n}$ is the hyperbolic space of constant sectional curvature $-1,$ $n\geq2,$ and if \[ \limsup_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0, \] where $S_{R}$ is a geodesic sphere of $\mathbb{H}^{n}$ centered at fixed point $o\in\mathbb{H}^{n}$ with radius $R,$ then $u$ is constant. Moreover, given $C>0,$ there is a bounded non-constant harmonic function $v\in C^{\infty }\left( \mathbb{H}^{n}\right) $ such that \[ \lim_{R\rightarrow\infty}\left( e^{R}\sup_{S_{R}}\left\Vert \nabla v\right\Vert \right) =C. \] The first part of the above result is a consequence of a more general theorem proved in the paper which asserts that if $G$ is a non compact Lie group with a left invariant metric, $u\in C^{1}\left( G\right) $ a solution of a left invariant PDE (that is, if $v$ is a solution of the PDE on a domain $Ω$ of $G$, the composition $v_{g}:=v\circ L_{g}$ of $v$ with a left translation $L_{g}:G\rightarrow G,$ $L_{g}\left( h\right) =gh,$ is also solution on $L_{g}^{-1}\left( Ω\right) $ for any $g\in G),$ the PDE satisfies the comparison principle and% \[ \limsup_{R\rightarrow\infty}\left( \sup_{g\in B_{R}}\left\Vert \operatorname*{Ad}\nolimits_{g}\right\Vert \sup_{S_{R}}\left\Vert \nabla u\right\Vert \right) =0, \] where $\operatorname*{Ad}\nolimits_{g}:\mathfrak{g}\rightarrow\mathfrak{g}$ is the adjoint map of $G$ and $\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is constant.

math.DG↗
arXiv2021

On the Size of Minimal Surfaces in $\mathbb{R}^4$

The Gauss map $g$ of a surface $Σ$ in $\mathbb{R}^4$ takes its values in the Grassmannian of oriented 2-planes of $\mathbb{R}^4$: $G^+(2,4)$. We give geometric criteria of stability for minimal surfaces in $\mathbb{R}^4$ in terms of $g$. We show in particular that if the spherical area of the Gauss map $|g(Σ)|$ of a minimal surface is smaller than $2π$ then the surface is stable by deformations which fix the boundary of the surface.This answers a question of Barbosa and Do Carmo in $\mathbb{R}^4$.

math.DG↗
arXiv2016

Lissajous-toric knots

A point in the $(N,q)$-torus knot in $\mathbb{R}^3$ goes $q$ times along a vertical circle while this circle rotates $N$ times around the vertical axis. In the Lissajous-toric knot $K(N,q,p)$, the point goes along a vertical Lissajous curve (parametrized by $t\mapsto(\sin(qt+ϕ),\cos(pt+ψ)))$ while this curve rotates $N$ times around the vertical axis. Such a knot has a natural braid representation $B_{N,q,p}$ which we investigate here. If $gcd(q,p)=1$, $K(N,q,p)$ is ribbon; if $gcd(q,p)=d>1$, $B_{N,q,p}$ is the $d$-th power of a braid which closes in a ribbon knot. We give an upper bound for the $4$-genus of $K(N,q,p)$ in the spirit of the genus of torus knots; we also give examples of $K(N,q,p)$'s which are trivial knots.

math.GT↗
arXiv2015

Complete minimal submanifolds of compact Lie groups

We give a new method for manufacturing complete minimal submanifolds of compact Lie groups and their homogeneous quotient spaces. For this we make use of harmonic morphisms and basic representation theory of Lie groups. We then apply our method to construct many examples of compact minimal submanifolds of the special unitary groups.

math.DG↗
arXiv2015

Lissajous and Fourier Knots

We prove that any knot of $\mathbb{R}^3$ is isotopic to a Fourier knot of type $(1,1,2)$ obtained by deformation of a Lissajous knot.

math.GT↗
arXiv2015

Desingularization of branch points of minimal surfaces in $\mathbb{R}^4$ (II)

We desingularize a branch point $p$ of a minimal disk $F_0(\mathbb{D})$ in $\mathbb{R}^4$ through immersions $F_t$'s which have only transverse double points and are branched covers of the plane tangent to $F_0(\mathbb{D})$ at $p$. If $F_0$ is a topological embedding and thus defines a knot in a sphere/cylinder around the branch point, the data of the double points of the $F_t$'s give us a braid representation of this knot as a product of bands.

math.DG↗
arXiv2013

On harmonic morphisms from 4-manifolds to Riemann surfaces and local almost Hermitian structures

We investigate the structure of a harmonic morphism $F$ from a Riemannian 4-manifold M^4 to a 2-surface $N^2$ near a critical point $m_0$. If $m_0$ is an isolated critical point or if $M^4$ is compact without boundary, we show that $F$ is pseudo-holomorphic w.r.t. an almost Hermitian structure defined in a neighbourhood of $m_0$. If $M^4$ is compact without boundary, the singular fibres of $F$ are branched minimal surfaces.

math.DG↗
arXiv2012

Some properties of simple minimal knots

A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the embedding of the disk in $\mathbb{C}^2$ are enough to determine the knot type, we talk of a simple minimal knot. Such a knot is given by three integers $N < p,q$; denoted by $K(N,p,q)$, it can be parametrized in the cylinder as $e^{iθ}\mapsto (e^{Niθ},\sin qθ,\cos pθ)$. From this expression stems a natural representation of $K(N,p,q)$ as an $N$-braid. In this paper, we give a formula for its writhe number, i.e. the signed number of crossing points of this braid and derive topological consequences. We also show that if $q$ and $p$ are not mutually prime, $K(N,p,q)$ is periodic. Simple minimal knots are a generalization of torus knots.

math.DG↗
arXiv2007

Singularity Knots of Minimal Surfaces in $\mathbb{R}^4$

We study knots in $\mathbb{S}^3$ obtained by the intersection of a minimal surface in $\mathbb{R}^4$ with a small 3-sphere centered at a branch point. We construct examples of new minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be an iterated knot of a minimal surface. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

math.DG↗
arXiv2007

Milnor numbers for 2-surfaces in 4-manifolds

In this paper (S_n) is a sequence of surfaces immersed in a 4-manifold which converges to a branched surface S_0. Up to sign, μ^T_p (resp. μ^N_p) will denote the amount of curvature of the tangent bundles TS_n (resp. the normal bundles NS_n) which concentrates around a singular point p of S_0 when n goes to infinity. By a slight abuse of notation, we call μ_p^T (resp. μ_p^N) the tangent (resp. normal) Milnor number of S_n at p. These numbers are not always well-defined; we discuss assumptions under which, if μ^T exists, then μ^N also exists and is smaller than -μ^T . When the second fundamental forms of the S_n's have a common L^2 bound, we relate μ^T and μ^N to a bubbling-off in the Grassmannian G_2^+(M).

math.DG↗
arXiv2007

Branched immersions and braids

Branch points of a real 2-surface S in a 4-manifold M generalize the branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when S is topologically embedded in M. It defines a link L(p), the components of which are closed braids with the same axis up to orientation. If S is closed without boundary, the contribution of p to the degree of the normal bundle of S in M can be computed on the link L(p) in terms of the algebraic crossing numbers of its components and on their linking numbers one with another.

math.DG↗