arXiv research

Classifies and constructs translators for curvature flows.

The study finds translators for higher order mean curvature flows in Euclidean and hyperbolic spaces.

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

The paper proves existence and classification of translating solitons in warped product manifolds.

Stability of catenoid in 4D Minkowski space proven without symmetry assumptions.

We classify all the translating solitons to the mean curvature flow in the three-dimensional Heisenberg group that are invariant under the action of some one-parameter group of isometries of the ambient manifold. The problem is solved considering any canonical deformation of the standard Riemannian metric of the Heisen…

Researchers classify translators and rotators in hyperbolic 3-space for mean curvature flow.

In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in $H^2 \times R$. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindelöf's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. …

Proves uniqueness of catenoid-like shapes in a ball.

In this note we construct a vase of catenoids - a symmetric immersed minimal surface with planar and catenoid ends.

Study on catenoid stability using asymmetric potentials.

This paper concerns some stability properties of higher dimensional catenoids in $\rr^{n+1}$ with $n\ge 3$. We prove that higher dimensional catenoids have index one. We use $δ$-stablity for minimal hypersurfaces and show that the catenoid is $\frac 2n$-stable and a complete $\frac 2n$-stable minimal hypersurface is a …

Catenoids in de Sitter $3$-space $S^3_1$ belong to a certain class of space-like constant mean curvature one surfaces. In a previous work, the authors classified such catenoids, and found that two different classes of countably many exceptional elliptic catenoids are not realized as closed subsets in $S^3_1$. Here we s…

Let $\mathcal{K}$ be the space of properly embedded minimal tori in quotients of $\R^3$ by two independent translations, with any fixed (even) number of parallel ends. After an appropriate normalization, we prove that $\mathcal{K}$ is a 3-dimensional real analytic manifold that reduces to the finite coverings of the ex…

The paper constructs surfaces with prescribed mean curvature in a specific space.

In 3-dimensional Lorentz-Minkowski space we determine the number of catenoids connecting two coaxial circles in parallel planes. This study is separated according to the types of circles and the causal character (spacelike and timelike) of the catenoid.

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this …

In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…

We describe a 3-parametric family $\mathcal{K}$ of properly embedded minimal tori with four parallel ends in quotients of $\mathbb{R}^3$ by two independent translations, which we will call the \textit{Standard Examples.} These surfaces generalize the examples given by Karcher, Meeks and Rosenberg in \cite{ka4,ka6,mr3}.…

The critical catenoid is uniquely determined by certain symmetries of its boundary.

Researchers create minimal surfaces with Scherk ends and find catenoid limits.

This paper studies non-compact Ricci surfaces with catenoidal ends.

Study calculates the renormalized area of catenoids in hyperbolic spaces.

We consider an appoximation of a catenoid constructed from "odd" truncated cones that maintains minimality in a certain sense. Thorough this procedure, we obtain a discrete curve approximating a catenary by exploiting the fact that it is the function that generates a catenoid. In this investigation, the theory of the G…

For a complete minimal surface in the Euclidean 3-space, the so-called flux vector corresponds to each end. The flux vectors are balanced, i.e., the sum of those over all ends are zero. Consider the following inverse problem: For each balanced n vectors, find an n-end catenoid which attains given vectors as flux. Here,…

We give a Weierstrass type representation for semi-discrete minimal surfaces in Euclidean 3-space. We then give explicit parametrizations of various smooth, semi-discrete and fully-discrete catenoids, determined from either variational or integrable systems principles. Finally, we state the shared properties that those…

We show that asymptotically Schwarzschildean 3-manifolds cannot contain minimal surfaces obtained by perturbative deformations of a Euclidean catenoid, no matter how small the ADM mass of the ambient space and how large the neck of the catenoid itself. Such an obstruction is sharply three-dimensional and ceases to hold…

We show that the rotationally symmetric free boundary minimal catenoid in the unit ball in $\Bbb{R}^3$ has Morse index equal to $4$.

In earlier work of NK new closed embedded smooth minimal surfaces in the round three-sphere $\mathbb{S}^3(1)$ were constructed, each resembling two parallel copies of the equatorial two-sphere $\mathbb{S}^2_{eq}$ joined by small catenoidal bridges, with the catenoidal bridges concentrating along two parallel circles, o…

We give a parameterization of Alfred Gray's Elliptical Catenoid and Elliptical Hellicoid using Jacobi's elliptic functions. This parameterization avoids some problems present in the original depiction of these surfaces.

Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.

This thesis constructs cmc 1/2 surfaces from catenoids, proving convergence and solving boundary value problems.

The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.

Solves constant mean curvature Dirichlet problem on catenoids with improved estimates.

Constructs minimal surfaces near the boundary of a ball.

We show that an embedded minimal annulus $Σ^2 \subset B^3$ which intersects $\partial B^3$ orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique…

The paper classifies stable free boundary minimal hypersurfaces outside a ball.

For each end of complete minimal surface in the Euclidean 3-space, the flux vector is defined. It is well-known that the sum of the flux vector over all ends are zero. Consider the following inverse problem: For each balanced n-vectors, find an n-end catenoid which realizes these vectors as flux. Here, an n-end catenoi…

In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Σ$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD sol…

In this note, we use a result of Osserman and Schiffer \cite{OS} to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To the best of our knowledge, this fact has gone unremarked upon in the literature. …

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus yet maintain the symmetry of their genus-zero counterparts constructed by Jorge…

The article constructs helicoidal and catenoidal minimal surfaces in a Lie group.

New method characterizes minimal surfaces in 3D space.

Constructs minimal immersions with singularities.

In this article, we show that the critical catenoid, as a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, has index $4$. We also prove that a free boundary minimal surface of the unit ball in $\mathbb{R}^3$, that is not a flat disk, has index at least $4$.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are …

Study of immersions with Willmore energy leading to spherical and catenoid bubbles.

The study explores special surfaces in a normed space.