arXiv subjects

J. P. dos Santos

Publications and source records attributed to J. P. dos Santos.

7 recordsLinked to original sources

arXiv2026

Half-space type theorems for a class of weighted minimal surfaces in $\mathbb{R}^{3}$

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic growth, we prove that there are no proper surfaces contained either in two transverse vertical half-spaces of $\mathbb{R}^3$ or in a half-space determined by a non-vertical plane. We show that this second result holds in a more general context, namely, for a class of stochastically complete weighted minimal surfaces. In this setup, we also prove a result for surfaces contained in regions bounded by cones. Furthermore, for stochastically complete weighted minimal surfaces satisfying restrictions on their principal curvatures, we establish a version of the classic strong half-space result due to Hoffman-Meeks.

math.DG↗

arXiv2025

Distributions with locally free tangent sheaf

In the paper Stability of Holomorphic Foliations with Split Tangent Sheaf one finds a study of the locus $\mathrm{Dec}$ where the tangent sheaf of a {\it family} of foliations in $\mathbb{P}_{\mathbb C}^n$ is {\it decomposable}, i.e. a sum of line bundles. A prime conclusion is an ``openness'' result: once the singular locus has sufficiently large codimension, $\mathrm{Dec}$ turns out to be open. In the present paper, we study the locus $\mathrm{LF}$ of points of a family of distributions where the tangent sheaf is {\it locally free}. Through general Commutative Algebra, we show that $\mathrm{LF}$ is open provided that singularities have codimension at least three. When dealing with foliations rather than distributions, the condition on the lower bound of the singular set can be weakened by the introduction of ``Kupka'' points. We apply the available ``openness'' results to families in $\mathbb{P}_{\mathbb C}^n$ and in $\mathcal B$, the variety of Borel subgroups of a simple group. By establishing a theorem putting in bijection irreducible components of the space of two-dimensional subalgebras of a given semi-simple Lie algebra and its nilpotent orbits, we conclude that the space of foliations of {\it rank two} on $\mathbb{P}_{\mathbb C}^n$ and $\mathcal B$, may have quite many irreducible components as $n$ and $\dim \mathcal B$ grow. We also set in place several algebro-geometric foundations for the theory of families of distributions in two appendices.

math.AG↗

arXiv2024

Solitons to Mean Curvature Flow in the hyperbolic 3-space

We consider {translators} (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic $3$-space $\mathbb H^3$, providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete rotational translators in $\mathbb H^3$, one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in $\mathbb H^3$ and apply it to prove that properly immersed translators to MCF in $\mathbb H^3$ are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the $x_3$-axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in $\mathbb H^3$ which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in $\mathbb H^3$ and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in $\mathbb R^3$.

math.DG↗

arXiv2024

Stationary surfaces of height-dependent weighted area functionals in $\mathbb{R}^3$ and $\mathbb{L}^3$

We describe a general correspondence between weighted minimal surfaces in $\mathbb{R}^3$ and weighted maximal surfaces with some admissible singularities in $\mathbb{L}^3$, for a class of functions $φ$ which provides the corresponding weight. For these families of surfaces, we provide a Weierstrass representation when $\dotφ\neq 0$ and analyze in detail the asymptotic behavior of both such a weighted maximal surface around its singular set and its corresponding weighted minimal immersion around the nodal set of its angle function, establishing criteria that allow us to easily determine the type of singularity and classify the associated moduli spaces.

math.DG↗

arXiv2021

Uniqueness of the $[φ,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour

We establish a uniqueness result for the $[φ,\vec{e}_{3}]$-catenary cylinders by their asymptotic behaviour. Well known examples of such cylinders are the grim reaper translating solitons for the mean curvature flow. For such solitons, F. Martín, J. Pérez-García, A. Savas-Halilaj and K. Smoczyk proved that, if $Σ$ is a properly embedded translating soliton with locally bounded genus, and $\mathcal{C}^{\infty}$-asymptotic to two vertical planes outside a cylinder, then $Σ$ must coincide with some grim reaper translating soliton. In this paper, applying the moving plane method of Alexandrov together with a strong maximum principle for elliptic operators, we increase the family of $[φ,\vec{e}_{3}]$-minimal graphs where these types of results hold under different assumption of asymptotic behaviour.

math.DG↗

arXiv2021

Hypersurfaces of Constant Higher Order Mean Curvature in $M\times\mathbb{R}$

We consider hypersurfaces of products $M\times\mathbb R$ with constant $r$-th mean curvature $H_r\ge 0$ (to be called $H_r$-hypersurfaces), where $M$ is an arbitrary Riemannian $n$-manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds $M,$ including all simply connected space forms and the hyperbolic spaces $\mathbb{H}_{\mathbb F}^m$ (rank $1$ symmetric spaces of noncompact type). We construct and classify complete rotational $H_r(\ge 0)$-hypersurfaces in $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ and in $\mathbb S^n\times\mathbb R$ as well. They include spheres, Delaunay-type annuli and, in the case of $\mathbb{H}_{\mathbb F}^m\times\mathbb R,$ entire graphs. We also construct and classify complete $H_r(\ge 0)$-hypersurfaces of $\mathbb{H}_{\mathbb F}^m\times\mathbb R$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett-Liebmann-type theorem by showing that a compact, connected and strictly convex $H_r$-hypersurface of $\mathbb H^n\times\mathbb R$ or $\mathbb S^n\times\mathbb R$ $(n\ge 3)$ is a rotational embedded sphere. Other uniqueness results for complete $H_r$-hypersurfaces of these ambient spaces are obtained.

math.DG↗

arXiv2020

Mean convex properly embedded $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$

We establish curvature estimates and a convexity result for mean convex properly embedded $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$, i.e., $φ$-minimal surfaces when $φ$ depends only on the third coordinate of $\mathbb{R}^3$. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana, for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in $\mathbb{R}^3$, we use a compactness argument to provide curvature estimates for a family of mean convex $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$. We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded $[φ,\vec{e}_{3}]$-minimal surface in $\mathbb{R}^{3}$ with non positive mean curvature when the growth at infinity of $φ$ is at most quadratic.

math.DG↗