arXiv research

The paper develops theory for holomorphic null curves in SL2(C).

Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.

In this paper we survey some recent contributions by the authors to the theory of null holomorphic curves in the complex Euclidean space $\mathbb C^3$, as well as their applications to null holomorphic curves in the special linear group $SL_2(\mathbb C)$, minimal surfaces in the Euclidean space $\mathbb R^3$, and const…

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${…

Develops explicit formulas for minimal immersions in 5D space.

We study pseudoholomorphic curves in the nearly Kalher $\mathbf{CP}^3$. It is shown that a class of curves called null-torsion are in one to one correspondence with the integrals of a holomorphic contact system on the usual Kahler $\mathbb{CP}^3$ studied by Bryant. Browing Bryant's result we get plenty of such curves. …

Quaternionic approach to conformal superminimal surfaces in four-space

We construct a complete bounded immersed null holomorphic curve in C^3, which is a recovery of the previous version of the paper Calc. Var. and PDE's vol 36 (2009); Erratum: to appear in Calc. Var. and PDE's, doi:10.1007/s00526-009-0226-5 by the last three authors on this subject.

In this paper we prove that every bordered Riemann surface M admits a complete proper null holomorphic embedding into a ball of the complex Euclidean $3$-space $\mathbb{C}^3$. The real part of such an embedding is a complete conformal minimal immersion $M\to \mathbb{R}^3$ with bounded image. For any such $M$ we also co…

The study proves a strong parametric h-principle for minimal surfaces.

We find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group O(n) in terms of free holomorphic data which lead to formulae for all such harmonic maps. Our work reveals an interesting correspondence between certain harmonic maps and the free Wei…

In this paper, we show that for every conformal minimal immersion $u:M\to \mathbb{R}^3$ from an open Riemann surface $M$ to $\mathbb{R}^3$ there exists a smooth isotopy $u_t:M\to\mathbb{R}^3$ $(t\in [0,1])$ of conformal minimal immersions, with $u_0=u$, such that $u_1$ is the real part of a holomorphic null curve $M\to…

We prove that for any open orientable surface $S$ of finite topology, there exist a Riemann surface $\mathcal{M},$ a relatively compact domain $M\subset\mathcal{M}$ and a continuous map $X:\bar{M}\to\mathbb{C}^3$ such that: $\mathcal{M}$ and $M$ are homeomorphic to $S,$ $\mathcal{M}-M$ and $\mathcal{M}-\bar{M}$ contain…

Let $M$ be an open Riemann surface. It was proved by Alarcón and Forstnerič (arXiv:1408.5315) that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve $M\to\mathbb C^3$. In this paper, we prove the following much stronger result in this direction: for any $n\geq …

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves i…

The abstract proves the existence of CMC-1 surfaces with any complex structure in hyperbolic space.

Study isotropic curves on complex quadric with geometric relations.

In this study, we define a family of null curves in Minkowski 3-space and called null similar curves. We obtain some properties of these special curves. We show that two null curves are null similar curves if and only if these curves form a null Bertrand pair. Moreover, we obtain that the family of null geodesics and n…

Minimal surfaces and curves can have singularities removed by isotopy.

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theor…

This paper studies ruled real hypersurfaces in indefinite complex projective space.

In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C^2. To construct su…

The study examines null curves in specific geometric manifolds and their properties.

In this study, non-null Frenet-Mannheim curves and non-null Weakened Mannheim curves are investigated in Minkowski 3-space. Some characterizations for this curves are obtained.

For any open orientable surface $M$ and convex domain $Ω\subset \mathbb{C}^3,$ there exists a Riemann surface $N$ homeomorphic to $M$ and a complete proper null curve $F:N\toΩ.$ This result follows from a general existence theorem with many applications. Among them, the followings: For any convex domain $Ω$ in $\mathbb…

Minimal surfaces in Heisenberg group have null curves and lines.

In this paper we study null Bertrand curves in $R_{1}^{4}$ under the assumption the curve has a Cartan frame. We show that if the derivative vectors of the null Cartan curve in $R_{1}^{4}$ is linearly independent, then this curve is not a Bertrand curve. Since then the already known notion of null Bertrand curves in $R…

In this paper, we study inextensible flows of partially null and pseudo null curves in E_1^4. We give neccessary and sufficent conditions for inextensible flows of partially null and pseudo null curves in E_1^4

Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in Minkowski 3-spa…

Expanding on my former work along with the more recent work of Kasuya and Takase, we demonstrate that for a given link $L \subset M$ which is null-homologous in $H_1(M)$ and for any smooth oriented 2-plane field $η$ over $L$ there exists a smooth embedding $F:M \hookrightarrow \mathbb{C}^3$ so that the set of complex t…

We study surfaces in TN that are area-stationary with respect to a neutral Kaehler metric constructed on TN from a riemannian metric g on N. We show that holomorphic curves in TN are area-stationary, while lagrangian surfaces that are area-stationary are also holomorphic and hence totally null. However, in general, are…

Geometric transformations on null curves in AdS induce KdV solutions.

Let $M$ be an open Riemann surface and $n\ge 3$ be an integer. We prove that on any closed discrete subset of $M$ one can prescribe the values of a conformal minimal immersion $M\to\mathbb{R}^n$. Our result also ensures jet-interpolation of given finite order, and hence, in particular, one may in addition prescribe the…

In this paper, we define the notion of eikonal helix and eikonal slant helix for null curves in the 4-dimensional Lorentzian manifold M 1 4 and give a characterization for the null curve to be the null eikonal helix. Moreover, we indicate an important relation between the null eikonal helix and null eikonal slant helix…

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ paralle…

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology.…

We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-g…

Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.

The local motion of a null curve in Minkowski 3-space induces an evolution equation for its Lorentz invariant curvature. Special motions are constructed whose induced evolution equations are the members of the KdV hierarchy. The null curves which move under the KdV flow without changing shape are proven to be the traje…

Generalizes Bryant's correspondence to gauge-theoretical settings.

We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski $3$-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the correspon…

Study of null φ-slant curves in specific 3D manifolds.

CMC-1 surfaces found on compact Riemann surfaces with Cantor sets.

First, we prove that indefinite Sasakian manifolds do not admit any screen conformal $r$-null submanifolds, tangent to the structure vector field. We, therefore, define a special class of null submanifolds, called; {\it contact screen conformal} $r$-null submanifold of indefinite Sasakian manifolds. Several characteriz…

In this paper, we investigate the similarity transformations in the Minkowski-n space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion. We determine all non-null self-similar curve…

Research shows curves in Walker 3-manifolds can lie in flat cylinders.

The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.

Paper connects surfaces in 4D and 3D spacetime.