arXiv research

New minimal surfaces derived from helicoids.

Study on helicoidal singular minimal surfaces with specific properties.

We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding hypersurface equations. Then we present a new two-parameter family of immersed helicoidal surfaces that rotate/translate with constant velocity under the flow. We look at…

There exists a properly embedded minimal surface of genus one with one end. The end is asymptotic to the end of the helicoid. This genus one helicoid is constructed as the limit of a continuous one-parameter family of screw-motion invariant minimal surfaces--also asymptotic to the helicoid--that have genus equal to one…

We study surfaces with constant anisotropic mean curvature which are invariant under a helicoidal motion. For functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation of Perdomo to the anisotropic case and show how all helicoidal constant anisotropic mean curvature s…

The paper analyzes self-similar solutions for mean curvature flow in 3D.

The paper extends Bour's theorem to helicoidal surfaces with singularities.

Using the lamination theory developed by Colding and Minicozzi for sequences of embedded, finite genus minimal surfaces with boundaries going to infinity \cite{CM5}, we show that the space of genus-one helicoids is compact (modulo rigid motions and homotheties). This generalizes a result of Hoffman and White \cite{HW}.

We develop a conservation law for constant mean curvature (CMC) surfaces introduced by Korevaar, Kusner and Solomon, and provide a converse, so as to characterize CMC surfaces by a conservation law. We work with `twizzler' construction, which applies a screw-motion to some base curve. We show that, excluding cylinders,…

Researchers classify and describe $K^α$-translators in Euclidean space.

Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out $L$ indpendent Clifford cones in ${\mathbb{R}}^{2N+2}$ via the multi-screw motion, we construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Also, we sweep out the $L$-rays Clifford cone (introduced in Sectio…

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

The class of traveling wave solutions of the sine-Gordon equation is known to be in 1-1 correspondence with the class of (necessarily singular) pseudospherical surfaces in Euclidean space with screw-motion symmetry: the pseudospherical helicoids. We explicitly describe all pseudospherical helicoids in terms of elliptic…

Classifies surfaces with zero mean curvature in a light cone.

Study helical motions of lines in 3D spaces, solving control problems.

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of i…

Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an ana…

The paper extends Bour's theorem to BCV spaces with constant mean curvature helicoids.

Study helicoidal surfaces with singular points using frontals.

Study helicoidal surfaces from frontals, revealing geometric rigidity and stability of singularities.

We prove by variational means the existence of a complete, properly embedded, genus-one minimal surface in R^3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid's axis, but that have infinitely many handles arranged periodically al…

We define a new kind of helicoidal surface of value m. A rotational surface which is isometric to the helicoidal surface of value m is revealed. In addition, we calculate some differential geometric properties of the helicoidal surface of value 3 in three dimensional Euclidean space.

In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains the singly periodic genus one helicoid of Hoffman, Karcher and Wei.

The paper studies timelike loxodromes on specific Lorentzian helicoidal surfaces.

We show that an embedded minimal disk in R^3 with large curvature is bilipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.

The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The objective of this paper is to give a uniqueness theorem for the singly periodic gen…

The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.

We prove that if a complete, properly embedded, finite-topology minimal surface in S^2 x R contains a line, then its ends are asymptotic to helicoids, and that if the surface is an annulus, it must be a helicoid.

For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while p…

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal s…

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Eu…

The height functions of K^(1/4)-flow translators in Euclidean space R^3 solve the unimodular Hessian equation. We explicitly and geometrically determine the moduli space of all helicoidal K^(1/4)-flow translators, which are generated from planar curves by the action of helicoidal groups.

Helicoidal surfaces rotate and translate under mean curvature flow.

New minimal surfaces found from vortex crystals.

The paper classifies solitons in the Heisenberg space.

Given an arbitrary $C^\infty$ Riemannian manifold $M^n$, we consider the problem of introducing and constructing minimal hypersurfaces in $M\times\mathbb{R}$ which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space $\mathbb{R}^3=\mathbb{R}^2\times\mathbb{R}$. Such hypersurfa…

The paper solves the Dirichlet problem for minimal surfaces on unbounded helicoidal domains.

In this note, we use the Lopez-Ros deformation introduced in [9] to show that any embedded genus-one helicoid must be symmetric with respect to rotation by 180 degrees around a normal line. This partially answers a conjecture of Bobenko from [3]. We also show this symmetry holds for an embedded genus-k helicoid $Σ$, pr…

We prove an existence result for helicoidal graphs with prescribed mean curvature in a large class of warped product spaces which comprises space forms.

Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.

There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one.

Colding and Minicozzi have shown that an embedded minimal disk $0\inΣ\subset B_R$ in $\Real^3$ with large curvature at 0 looks like a helicoid on the scale of $R$. Near 0, this can be sharpened: on the scale of $|A|^{-1}(0)$, $Σ$ is close, in a Lipschitz sense, to a piece of a helicoid. We use surfaces constructed by C…

The paper classifies helicoidal surfaces with specific curvature functions.

The paper classifies surfaces in the Heisenberg space invariant under specific isometries.

The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The resu…

We prove that for each positive integer g, there exists a complete minimal surface of genus g that is properly embedded in three-dimensional euclidean space and that is asymptotic to the helicoid.

In a previous paper the author introduced the notion of TreadmillSled of a curve, which is an operator that takes regular curves in R^2 to curves in R^2. This operator turned out to be very useful to describe helicoidal surfaces, for example, it provides an interpretation for the profile curve of helicoidal surfaces wi…