arXiv subjects

Carlos Peñafiel

Publications and source records attributed to Carlos Peñafiel.

6 recordsLinked to original sources

arXiv2025

Height estimates for surfaces with some constant curvature in $\mathbb{r} \times_{f} \mathbb{r}^{2}$

In this paper, we obtain the necessary equations in a conformal parameter induced by the first or second fundamental forms for a surface that is isometrically immersed in the warped product $\mathbb{R} \times_{f} \mathbb{M}^{2}(κ)$ where $\mathbb{M}^{2}(κ)$ denotes the complete, connected, simply connected, two-dimensional space form of constant curvature. The surface we will consider has either positive extrinsic curvature or positive mean curvature. In each case, we carry out some geometric applications to the theory of constant curvature surfaces immersed in $\mathbb{R} \times_{f} \mathbb{R}^{2}$ under certain conditions on the warping function $f$. Specifically, we derive height estimates for graph-type surfaces with either positive constant extrinsic curvature or positive constant mean curvature. In particular, we classify compact minimal graphs in such warped products. This article extends previous work on the study of constant curvature surfaces immersed in product spaces using conformal parameters, as well as the height estimates for constant curvature surfaces in the warped product $\mathbb{R} \times_{f} \mathbb{R}^{2}$.

math.DG↗

arXiv2024

A new characterization for Clifford hypersurfaces

For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $σ_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $σ_k \ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $σ_2 \ge 16$. When $n\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\ge 2$, there exists a positive constant $δ_k(n)<n$, if $|A|^2\ge δ_k(n)$, we have $σ_k\ge n^k$. All the equality holds iff $M$ is isometric to a Clifford hypersurface.

math.DG↗

arXiv2023

Elliptic Weingarten surfaces of minimal type in $\mathbb{R} \times_{h} \mathbb{R}$

In this paper, we study the elliptic Weingarten surfaces of minimal type immersed in the warped product space $\mathbb{R} \times_{h} \mathbb{R}$, when $h$ is a $C^{1}$-function in $\mathbb{R}^{2}$ with radial symmetry. That is, surfaces whose mean curvature $H$ and extrinsic curvature $K$ satisfy a relationship $H=f(H^{2}-K)$ where $f \in C^{1}(-ε,+\infty)$ with $ε> 0$, $f(0)=0$ and $4t(f'(t))^{2} < 1$ for $t \in (-ε,\infty)$. We show, under some assumptions about the warping function $h$, the existence and uniqueness of the rotationally-invariant examples of elliptic Weingarten of minimal type surfaces immersed in $\mathbb{R} \times_{h} \mathbb{R}$ as well as we study the geometric behavior of its generating curve.

math.DG↗

arXiv2020

On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped product $\mathbb{M}(κ)_f\times I$ (here, $\mathbb{M}(κ)$ denotes the 2-dimensional space form, having constant curvature $κ$, $I$ an interval and $f$ the warping function) is invariant by an one-parameter group of isometries of the ambient space. We also find the first integral of the ordinary differential equation that the profile curve satisfies (we mean, the curve which generates a invariant totally umbilical surface). Moreover, we construct explicit examples of totally umbilical surfaces, invariant by one-parameter group of isometries of the ambient space, by considering certain non-trivial warping function.

math.DG↗

arXiv2018

Height estimates for $H$-surfaces in the warped product $\mathbb{M}\times_f\mathbb{R}$

In this article, we consider compact surfaces $Σ$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $Γ=\partialΣ\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the warped product $ \mathbb{M}\times_f\mathbb{R} $, here $ \mathbb{M} $ denotes a Hadamard surface. We obtain height estimate for a such surface $Σ$ having positive constant mean curvature involving the area of a part of $Σ$ above of $ \mathbb{M} _0$ and the volume it bounds. Also we give general conditions for the existence of rotationally-invariant topological spheres having positive constant mean curvature $H$ in the warped product $\mathbb{H}\times_f\mathbb{R}$, where $\mathbb{H}$ denotes the hyperbolic disc. Finally we present a non-trivial example of such spheres.

math.DG↗

arXiv2015

Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$

In this article, we study complete surfaces $Σ$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $Σ$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b>0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.

math.DG↗