arXiv research

The paper identifies a new geometric and spectral phenomenon in the critical hyperbolic catenoid family.

Proves uniqueness of catenoid-like shapes in a ball.

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. …

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

Critical spherical catenoids have Robin nullity and asymptotic radius determined.

Study calculates the renormalized area of catenoids in hyperbolic spaces.

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

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…

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

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

Stability of catenoid in hyperbolic space proven without symmetry assumptions.

Constructs minimal annuli with free boundary in hyperbolic 3-space.

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 $4$.

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stabl…

We investigate the close relationship between minimal surfaces in Euclidean 3-space and constant mean curvature 1 surfaces in hyperbolic 3-space. Just as in the case of minimal surfaces in Euclidean 3-space, the only complete connected embedded constant mean curvature 1 surfaces with two ends in hyperbolic space are we…

New method characterizes minimal surfaces in 3D space.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

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 …

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 …

For a family of spherical minimal catenoids C_a in the hyperbolic 3-space, there exist two constants 0<a_c<a_l such that the following are true: (1) C_a is an unstable minimal surface with index one if a<a_c, (2) C_a is a stable minimal surface if a>=a_c, and (3) C_a is a least area minimal surface in the sense of Meek…

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

Minimal surfaces reflect across spheres, proving annulus uniqueness.

New minimal annuli found in unit ball, solving old problems.

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…

Analyzes a critical spherical catenoid in hyperbolic space, proving its index and nullity.

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…

In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$ such that $\bullet$ $\mathcal{C}_a$ is an unstable minimal surface with index o…

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

In this note we investigate free boundary minimal surfaces in the Euclidean 3-space, and by using holomorphic techniques developed by Fraser and Schoen we prove that the free boundary minimal annulus is the critical catenoid.

We show existence of constant mean curvature 1 surfaces in both hyperbolic 3-space and de Sitter 3-space with two complete embedded ends and any positive genus up to genus twenty. We also find another such family of surfaces in de Sitter 3-space, but with a different non-embedded end behavior.

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

We prove that the only embedded free boundary minimal surface $Σ$ in $\mathbb{B}^3$ with index $4$ is the critical catenoid. This extends fundamental work of A. Fraser and R. Schoen, as well as the work of H. Tran.

We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss curvature. We provide explicit parametrizations and deduce existence of complete hyperbolic linear Weingarten surfaces.

We show that, among free boundary minimal surfaces in the unit ball in the three-dimensional Euclidean space, the flat equatorial disk and the critical catenoid are characterised by a pinching condition on the length of their second fundamental form.

We present a global representation for surfaces in 3-dimensional hyperbolic space with constant mean curvature 1 (CMC-1 surfaces) in terms of holomorphic spinors. This is a modification of Bryant's representation. It is used to derive explicit formulas in hypergeometric functions for CMC-1 surfaces of genus 0 with thre…

We extend to higher codimension earlier characterization of the equatorial disk and the critical catenoid by a pinching condition on the length of their second fundamental form among free boundary minimal surfaces in the three dimensional Euclidean ball due to L. Ambrozio and I. Nunes.

For all $n$, we define the $n$-dimensional critical catenoid $M_n$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $\Bbb{R}^{n+1}$. We show that the Morse index $\text{MI}(n)$ of $M_n$ satisfies the following asymptotic estimate as …

In this paper, we study the critical case of the Allard regularity theorem. Combining with Reifenberg's topological disk theorem, we get a critical Allard-Reifenberg type regularity theorem. As a main result, we get the topological finiteness for a class of properly immersed surfaces in $\mathbb{R}^n$ with finite Willm…

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue sigma_1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Sigma with genus gamma and k boundary components we obtain the upper bound sigma_1L(\partial Σ) \leq 2(2gamma+k)π. We atte…

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…

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

Proves the index of a Möbius band in 4D ball equals 5.

We derive necessary conditions on the parameters of the ends of a CMC-1 trinoid in hyperbolic 3-space $H^{3}$ with symmetry plane by passing to its conjugate minimal surface. Together with Daniel's results, this yields a classification of generic symmetric trinoids. We also discuss the relation to other classification …

Constructs a unique surface in a ball with specific properties.

We develop a theory of "minimal $θ$-graphs" and characterize the behavior of limit laminations of such surfaces, including an understanding of their limit leaves and their curvature blow-up sets. We use this to prove that it is possible to realize families of catenoids in euclidean space as limit leaves of sequences of…