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A. L. Martínez-Triviño

Publications and source records attributed to A. L. Martínez-Triviño.

9 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

On the first Neumann eigenvalue for critical points of a weighted area functional with asymptotically flat ends

In the following work, we obtain a lower bound for the first Neumann eingevalue of the drift Laplacian $Δ^φ$ for a family of properly embedded $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^3$ with concave function $φ$ and asymptotically flat ends. As application, we obtain a control on the topology for that surfaces with finite total curvature. The strategy will consists in an integration of the Bouchner's formula by the works of A. Lichnerowicz and S. Brendle, R. Tsiamis and relate said eigenvalue with the Poincare's constant.

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↗
arXiv2023

Calabi-Bernstein type results for critical points of a weighted area functional in $\mathbb{R}^{3}$ and $\mathbb{L}^{3}$

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete $[φ,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$ whose Gauss maps lie on compacts subsets of open hemispheres of $\mathbb{S}^{2}$. We also give a general non-existence result for complete spacelike $[φ,\vec{e}_{3}]$-maximal surfaces in $\mathbb{L}^{3}$ and, in particular, we obtain a Calabi-Bernstein type result when $\dotφ$ is bounded.

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↗
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↗
arXiv2020

Equilibrium of Surfaces in a Vertical Force Field

In this paper we study $φ$-minimal surfaces in $\mathbb{R}^3$ when the function $φ$ is invariant under a two-parametric group of translations. Particularly those which are complete graphs over domains in $\mathbb{R}^2$. We describe a full classification of complete flat embedded $φ$-minimal surfaces if $φ$ is strictly monotone and characterize rotational $φ$-minimal surfaces by its behavior at infinity when $φ$ has a quadratic growth.

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