arXiv research

Schwartz's solution to the Björling problem leads to an equivalence class of spatial strips S(t)=(c(t),n(t)) which produce equivalent minimal surfaces. For the particular case when the generating strip S(t) belongs to some plane E and c(t) is symmetric with respect to some straight line in E, the symmetries of the mini…

Study delta invariant of minimal generic curves on rational surfaces.

In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space $R_{n}$.

Sharp proof of sub-Riemannian length-minimizing curves being at least $C^2$

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

Optimal flat ribbons can be created from nonplanar curves with minimal energy.

A flat virtual link is a finite collection of oriented closed curves $\mathfrak L$ on an oriented surface $M$ considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves $(L_1,L_2)$, we show that the minimal number of intersecti…

Characterizes curves for minimal surfaces in de Sitter space.

The paper analyzes discrete approximations to minimize curve length in Euclidean space.

We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hami…

Paper proves unique energy-minimizing curves in constrained spaces.

Proves prime theta-curves for knots on minimal genus surfaces.

The paper proves existence of minimal homotopies for immersed planar curves.

We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.

Paper finds minimal number of curves in surface systems.

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …

The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.

The paper finds local minimizers for obstacle avoidance on curved spaces.

Generalizes embeddedness result for extreme curves.

The paper finds curves minimizing elastic energy pinned at endpoints.

Survey on minimal rational curves and their geometric structures.

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…

Solves area-minimizing surface problem for finite curves in H^2xR.

Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.

Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…

New minimal tori found in curved spaces.

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz reflection extends to a complete minimal surface. Studying the group of Schwarz refl…

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call {\it quasi-$k$-curved metrics}. Quasi-$k$-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity.

Study of rational curves in complex manifolds with specific normal bundles.

Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.

Minimal cylinders in Heisenberg group characterized using loop group method.

Unified approach classifies stable and minimal elastic curves.

In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asympt…

Study minimizes crossing points of up to 12 curves on a genus 2 surface.

We prove that a normal vector field along a curve in R3 is rotation minimizing (RM) if and only if it is parallel respect to the normal connection. This allows us to generalize all the results of RM vectors and frames to curves immersed in Riemannian manifolds.

We investigate minimal surfaces passing a given curve in $R^{3}$. Using the Frenet frame of a given curve and isothermal parameter, we derive the necessary and sufficient condition for minimal surface. Also we derive the parametric representation of two minimal surface families passing a circle and a helix as examples.

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

New findings on hypersurfaces in Euclidean space that are both maximal and minimal.

This paper is about interpolating minimal surfaces between two real analytic curves, a and b, each of which are simple real analytic curves, using the Björling-Schwarz formula in the domain where it is valid, changing the normal distributions on inital curves. We insert curves $l_1,...l_L$ at specific locations and cla…

A translation surface of Euclidean space $\r^3$ is the sum of two regular curves $α$ and $β$, called the generating curves. In this paper we classify the minimal translation surfaces of $\r^3$ and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is pro…

Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we s…

Study transverse $J$-holomorphic curves linking nearly Kähler $\mathbb{CP}^3$ to minimal surfaces.

The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.

Characterizes minimizing curves in Riemannian manifolds.

The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.

Minimal surfaces in Heisenberg group have null curves and lines.

The paper proves properties of curves in Riemannian manifolds.