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Jaime Ripoll

Publications and source records attributed to Jaime Ripoll.

At least 19 recordsLinked to original sources
arXiv2026

Artificial Intelligence and the Autonomization of Mathematics

This essay examines the relationship between artificial intelligence and the historical evolution of modern Mathematics. Rather than viewing AI as an external rupture, we argue that its effectiveness reveals a structural tendency already present in the autonomization of Mathematics itself. Modern Mathematics progressively developed formal environments that became increasingly autonomous, internally stable, and structurally navigable, reducing their dependence on concrete experience. In this context, the affinity between AI and contemporary mathematical practice appears less accidental than it may initially seem. The essay also discusses possible limits of formal navigability, particularly regarding the emergence of genuinely new conceptual regimes and forms of mathematical intelligibility. Husserl's reflections on mathematization and the distancing of science from the Lebenswelt provide a broader philosophical framework for understanding this process. We finally suggest that the contemporary debate on AI may concern less a threat to Mathematics itself than a challenge to the historical image of the mathematician as the privileged interpreter of mathematical structures.

math.GM↗
arXiv2025

An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary differential equation, which allows the construction of explicit boundary barriers and it can also be applied to unbounded domains. In the setting of Hadamard manifolds, this condition relates the geometry of the domain to the admissible boundary data in a direct way. In Euclidean space, the condition leads to solvability under geometric hypotheses of a different nature from those in the classical Jenkins-Serrin theory, and in some configurations it applies where the Jenkins-Serrin method does not. A central point of the present approach is that the geometric restrictions and the boundary data enter independently. The same barrier construction can be used for graphical mean curvature flow. This yields short-time existence with prescribed boundary values even when the boundary of the domain is not mean convex. When mean convexity is present, one recovers the classical graphical setting.

math.AP↗
arXiv2025

A lower bound of the energy functional of a class of vector fields and a characterization of the sphere

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $\Xi (M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert ^{2}dM}{\int_{M}\left\Vert X\right\Vert ^{2}dM}, X\in \Xi (M)\backslash\{0\}. % \] Let $G\in\operatorname*{Iso}\left(M\right)$ be a compact Lie subgroup of the isometry group of $M$ acting with cohomogeneity $1$ on $M$. Assume that any isotropy subgroup of $G$ is non trivial and acts with no fixed points on the tangent spaces of $M$, except at the null vectors. We prove in this note that under these hypothesis, if the Ricci curvature $\operatorname*{Ric}\nolimits_{M}$ of $M$ has the lower bound $\operatorname*{Ric}\nolimits_{M}\geq(n-1)k^{2}$, then $\operatorname*{F(X)}\geq(n-1)k^{2}$, for any $G$-invariant vector field $X\in \Xi (M)\backslash\{0\}$, and the equality occurs if and only if $M$ is isometric to the n-dimensional sphere $\mathbb{S}^{n}_k$ of constant sectional curvature $k^{2}$. In this case $X$ is an infimum of $F$ on $\Xi (\mathbb{S}^{n}_k).$

math.DG↗
arXiv2023

The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of $\mathbb{R}^{m}$

We consider a helicoidal group $G$ in $\mathbb{R}^{n+1}$ and unbounded $G$-invariant $C^{2,α}$-domains $Ω\subset\mathbb{R}^{n+1}$ whose helicoidal projections are exterior domains in $\mathbb{R}^{n}$, $n\geq2$. We show that for all $s\in\mathbb{R}$, there exists a $G$-invariant solution $u_{s}\in C^{2,α}\left( \overlineΩ\right) $ of the Dirichlet problem for the minimal surface equation with zero boundary data which satisfies $\sup_{\partialΩ}\left\vert \operatorname{grad}u_{s}\right\vert =\left\vert s\right\vert $. Additionally, we provide further information on the behavior of these solutions at infinity.

math.DG↗
arXiv2021

Group invariant solutions of certain partial differential equations

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla u\right\Vert \right) }{\left\Vert \nabla u\right\Vert }\nabla u\right) & =0\text{ in }Ω\\ u|\partialΩ& =φ\end{align*} where $Ω$ is a $G-$invariant domain of $C^{2,α}$ class in $M$ and $φ\in C^{0}\left( \partial\overlineΩ\right) $ a $G-$invariant function. Two classical PDE's are included in this family: the $p-$Laplacian $(a(s)=s^{p-1},$ $p>1)$ and the minimal surface equation $(a(s)=s/\sqrt {1+s^{2}}).$ Our motivation is to present a method in studying $G$-invariant solutions for noncompact Lie groups which allows the reduction of the Dirichlet problem on unbounded domains to one on bounded domains.

math.DG↗
arXiv2021

On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Given an exterior domain $Ω$ with $C^{2,α}$ boundary in $\mathbb{R}^{n}$, $n\geq3$, we obtain a $1$-parameter family $u_γ\in C^{\infty}\left(Ω\right) $, $\left\vert γ\right\vert \leqπ/2$, of solutions of the minimal surface equation such that, if $\left\vert γ\right\vert <π/2$, $u_γ\in C^{\infty}\left( Ω\right) \cap C^{2,α}\left( \overlineΩ\right) $, $u_γ|_{\partialΩ}=0$ with $\max_{\partialΩ}\left\Vert \nabla u_γ\right\Vert =\tanγ$ and, if $\left\vert γ\right\vert =π/2$, the graph of $u_γ$ is contained in a $C^{1,1}$ manifold $M_γ\subset\overlineΩ\times\mathbb{R}$ with $\partial M_γ=\partialΩ$. Each of these functions is bounded and asymptotic to a constant \[ c_γ=\lim_{\left\Vert x\right\Vert \rightarrow\infty}u_γ\left( x\right) . \] The mappings $γ\rightarrow u_γ\left( x\right) $ (for fixed $x\inΩ$) and $γ\rightarrow c_γ$ are strictly increasing and bounded. The graphs of these functions foliate the open subset of $\mathbb{R}^{n+1}$ \[ \left\{ \left( x,z\right) \inΩ\times\mathbb{R}\text{, }-u_{π/2}\left( x\right) <z<u_{π/2}\left( x\right) \right\} . \] Moreover, if $\mathbb{R}^{n}\backslashΩ$ satisfies the interior sphere condition of maximal radius $ρ$ and if $\partialΩ$ is contained in a ball of minimal radius $\varrho$, then \[ \left[ 0,σ_{n}ρ\right] \subset\left[ 0,c_{π/2}\right] \subset\left[ 0,σ_{n}\varrho\right] , \] where \[ σ_{n}=\int_{1}^{\infty}\frac{dt}{\sqrt{t^{2\left( n-1\right) }-1}}. \] One of the above inclusions is an equality if and only if $ρ=\varrho$, $Ω$ is the exterior of a ball of radius $ρ$ and the solutions are radial.

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↗
arXiv2019

Convexity at infinity in Cartan-Hadamard manifolds and applications to the asymptotic Dirichlet and Plateau problems

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any prescribed rate. These improve previous results of Anderson, Borbély, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with Z_2-multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded (k-1)-dimensional submanifold of \partial_{\infty}M, 2\leq k\leq n-1, as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with Z-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to -infinity) of the curvature.

math.DG↗
arXiv2019

Asymptotic and exterior Dirichlet problems for the minimal surface equation in the Heisenberg group with a balanced metric

It is proved that the Heisenberg group $\operatorname*{Nil}\nolimits_{3}$ with a balanced metric, the sum of the left and right invariant metrics, splits as a Riemannian product $\mathbb{T\times Z}$, where $\mathbb{T}$ is a totally geodesic surface and $\mathbb{Z}$ the center of $\operatorname*{Nil}% \nolimits_{3}.$ It is then proved the existence of complete properly embedded minimal surfaces in $\operatorname*{Nil}\nolimits_{3}$ by solving the asymptotic Dirichlet problem for the minimal surface equation on $\mathbb{T}$. It is also proved the existence of complete properly embedded minimal surfaces foliating an open set of $\operatorname*{Nil}\nolimits_{3}$ having as boundary a given curve $Γ$ in $\mathbb{T},$ satisfying the exterior circle condition, by solving the exterior Dirichlet problem for the minimal surface equation in the unbounded connected component of $\mathbb{T}\backslashΓ$.

math.DG↗
arXiv2019

Gauss map and the topology of constant mean curvature hypersurfaces of $\mathbb{S}^{7}$ and $\mathbb{CP}^{3}$

We define a Gauss map $γ:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $γ$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $γ$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $\mathbb{CP}^{3}$ and obtain similar results for CMC hypersurfaces of this space.

math.DG↗
arXiv2018

Topological rigidity for closed hypersurfaces of elliptic space forms

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

math.DG↗
arXiv2018

Minimal isoparametric submanifolds of $\mathbb{S}^{7}$ and octonionic eigenmaps

We use the octonionic multiplication $\cdot$ of $\mathbb{S}^{7}$ to associate, to each unit normal section $η$ of a submanifold $M$ of $\mathbb{S}^{7},$ an octonionic Gauss map $γ_η:M\rightarrow\mathbb{S}^{6},$ $γ_η(x)=x^{-1}\cdotη(x),$ $x\in M,$ where $\mathbb{S}^{6}$ is the unit sphere of $T_{1}\mathbb{S}^{7},$ $1$ is the neutral element of $\cdot$ in $\mathbb{S}^{7}.$ Denoting by $\mathcal{N}(M)$ the vector bundle of normal sections of $M$ we set, for $η$ $\in\mathcal{N}(M),$ $S_η(X)=-\left(\nabla_{X}η\right) ^{\top},$ $X\in TM.$ Considering the Hilbert-Schmidt inner product on the vector bundle $\mathcal{S}(M)=\left\{S_η, \ \text{}η\in\mathcal{N}(M)\right\} $ and defining the bundle map $\mathcal{B} :\mathcal{N}(M)\rightarrow\mathcal{S}(M)$ by $\mathcal{B}(η)=S_η,$ we prove that if $M$ is a minimal submanifold of $\mathbb{S}^{7}$ and $η\in\mathcal{N}(M)$ is unitary and parallel on the normal connection, then $γ_η$ is harmonic if and only if $η$ is an eigenvector of $\mathcal{B}^{\ast}\mathcal{B}:\mathcal{N}(M)\rightarrow\mathcal{N}(M),$ where $\mathcal{B}^{\ast}$ is the adjoint of $\mathcal{B}.$ If $M$ is an isoparametric compact minimal submanifold of codimension $k$ of $\mathbb{S}% ^{7}$ then $\mathcal{B}^{\ast}\mathcal{B}$ has constant non negative eigenvalues $0\leqσ_{1}\leq\cdots\leqσ_{k}$ and the associated eigenvectors $η_{1},\cdots,η_{k}$ form an orthonormal basis of $\mathcal{N}(M)$, parallel on the normal connection, such that each $γ_{η_{j}}$ is an eigenmap of $M$ with eigenvalue $7-k+$ $σ_{j}.$ Moreover, $σ_{j}=\Vert S_{η_{j}}\Vert^{2},$ $1\leq j\leq k.$

math.DG↗
arXiv2018

Notes on the Dirichlet problem of a class of second order elliptic partial differential equations on a Riemannian manifold

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_Ωϕ\left( \left\vert \nabla u\right\vert \right) , \] where $Ω$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $Ω=\mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($ϕ(s)=s^{p}$, $p>1)$ and the minimal surface equation ($ϕ(s)=\sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculus of Variations which seems to be common in the case of the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$ estimates of smooth solutions of the Euler-Lagrange equation. These estimates are obtained by a coordinate free calculus. Degenerate elliptic equations like the $p-$Laplacian are dealt with by an approximation argument. These notes address mainly researchers and graduate students interested in elliptic partial differential equations on Riemannian manifolds and may serve as a material for corresponding courses and seminars.

math.DG↗
arXiv2017

On the critical points of the energy functional on vector fields of a Riemannian manifold

Given a compact Lie subgroup $G$ of the isometry group of a compact Riemannian manifold $M$ with a Riemannian connection $\nabla,$ it is introduced a $G-$symmetrization process of a vector field of $M$ and it is proved that the critical points of the energy functional \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert ^{2}dM}{\int_{M}\left\Vert X\right\Vert ^{2}dM}% \] on the space of $\ G-$invariant vector fields are critical points of $F$ on the space of all vector fields of $M,$ and that this inclusion may be strict in general. One proves that the infimum of $F$ on $\mathbb{S}^{3}$ is not assumed by a $\mathbb{S}^{3}-$invariant vector field. It is proved that the infimum of $F$ on a sphere $\mathbb{S}^{n},$ $n\geq2,$ of radius $1/k,$ is $k^{2},$ and is assumed by a vector field invariant by the isotropy subgroup of the isometry group of $\mathbb{S}^{n}$ at any given point of $\mathbb{S}% ^{n}.$ It is proved that if $G$ is a compact Lie subgroup of the isometry group of a compact rank $1$ symmetric space $M$ which leaves pointwise fixed a totally geodesic submanifold of dimension bigger than or equal to $1$ then all the critical points of $F$ are assumed by a $G-$invariant vector field. Finally, it is obtained a characterization of the spheres by proving that on a certain class of Riemannian compact manifolds $M$ that contains rotationally symmetric manifolds and rank $1$ symmetric spaces$,$ with positive Ricci curvature $\operatorname*{Ric}\nolimits_{M}$, $F$ has the lower bound $\operatorname*{Ric}\nolimits_{M}/\left( n-1\right) $ among the $G-$ invariant vector fields, where $G$ is the isotropy subgroup of the isometry group of $M$ at a point of $M,$ and that his lower bound is attained if and only if $M$ is a sphere of radius $1/\sqrt{\operatorname*{Ric}\nolimits_{M}}.$

math.DG↗
arXiv2015

Complete minimal discs in Hadamard manifolds

Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.

math.DG↗