Paper proves unique energy-minimizing curves in constrained spaces.
problem Uniqueness of energy-minimizing curves in constrained spaces.
method Investigated energy-minimizing curves with fixed endpoints in a constrained space.
result Proved that the set of points for which the energy-minimizing curve is not unique has no interior points.
Proves prime theta-curves for knots on minimal genus surfaces.
problem Prime knots and essential arcs on Seifert surfaces.
method Analyzes prime knot unions with essential arcs on minimal genus surfaces.
result Each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.
The paper proves existence of minimal homotopies for immersed planar curves.
problem Existence of area-minimizing homotopies between homotopic curves in the plane.
method Geometric and variational approach, lifting curves into higher co-dimension, applying Douglas's solution of the Plateau problem.
result Uniform convergence of Douglas minimizers and minimal homotopy area minimization.
Paper finds minimal number of curves in surface systems.
problem Determining minimal number of curves in surface systems.
method Analyzes oriented surfaces of genus g for positive integers k and g.
result Exact minimal number of curves in filling k-systems.
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
Solves area-minimizing surface problem for finite curves in H^2xR.
problem Asymptotic Plateau problem for area-minimizing surfaces.
method Complete solution for finite curves in $\BHH$.
result Fairly complete solution for finite curves in $\BHH$.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for λ above which minimizers touch the obstacle, regardless of obstacle shape. Study minimizes crossing points of up to 12 curves on a genus 2 surface.
problem Minimizing intersection points of curves on a surface.
method Analyzes systems of up to 12 simple closed curves on a genus 2 surface to find the minimum crossing number.
result Determines the minimal crossing number of up to 12 curves on a genus 2 surface and proves the minimization systems are unique.
We investigate minimal surfaces passing a given curve in R3. 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.
Sharp proof of sub-Riemannian length-minimizing curves being at least C2
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving C2 regularity result Theorem 1.1 in [6] is sharp
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate O(N−1/2). 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 l1,...lL at specific locations and cla…
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…
Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Optimal flat ribbons can be created from nonplanar curves with minimal energy.
problem Finding the optimal shape of flat ribbons from nonplanar curves.
method Direct method of the calculus of variations.
result Optimal flat ribbons can be created with minimal bending energy, but they may have isolated planar points.
Study transverse J-holomorphic curves linking nearly Kähler CP3 to minimal surfaces.
problem Understanding J-holomorphic curves in nearly Kähler CP3. method Introducing transverse J-holomorphic curves and establishing Bonnet-type theorems. result Classification of flat tori and construction of moment-type maps.
Characterizes minimizing curves in Riemannian manifolds.
problem Finding optimal paths in curved spaces.
method Characterization of prox-regular sets and tangent cones.
result Necessary condition for minimizing curves in prox-regular sets.
Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.
problem Investigating rigidity and deformability of pseudoholomorphic curves in S6. method Analyzing moduli space of minimal surfaces isometric to pseudoholomorphic curves.
result Describes the moduli space of noncongruent minimal surfaces isometric to pseudoholomorphic curves.
Minimal surfaces in Heisenberg group have null curves and lines.
problem Characterizing timelike minimal surfaces in the Heisenberg group.
method Characterization through null curves and lines with prescribed curvatures.
result Minimal surfaces are defined by the multiplication of null curves and affine null lines.
The paper proves properties of curves in Riemannian manifolds.
problem Characterizing curves in Riemannian manifolds.
method Analyzing locally minimizing and weak geodesics.
result Locally minimizing curves are weak geodesics under certain conditions.
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 Rn.
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…
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…
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
Paper connects surfaces in 4D and 3D spacetime.
problem Finding relations between Lorentz surfaces in different spacetime dimensions.
method Weierstrass-type representations for null curves and surfaces.
result Relation between minimal Lorentz surfaces in R24 and R13. Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…
In this paper, Legendre curves on unit tangent bundle are given using rotation minimizing (RM) vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classifed.
A flat virtual link is a finite collection of oriented closed curves 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 (L1,L2), we show that the minimal number of intersecti…
New parametrizations for minimal timelike surfaces discovered.
problem Finding parametrizations for minimal timelike surfaces in specific spaces.
method Derived representation formulas for null curves leading to parametrizations of minimal timelike surfaces.
result Examples of minimal timelike surfaces constructed.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…
Study bounds index of minimal hypersurfaces in curved spaces.
problem Bounding the index of minimal hypersurfaces.
method Proved linear index bound using first Betti number and curvature.
result Index is bounded below by a linear function of first Betti number.
The Mittag-Leffler theorem is extended to meromorphic curves and minimal surfaces.
problem Extending the Mittag-Leffler theorem to meromorphic curves and minimal surfaces.
method Established a Mittag-Leffler-type theorem for meromorphic curves and minimal immersions, including interpolation and approximation.
result Complete minimal ends in R^5 are generically embedded, and open Riemann surfaces are characterized for minimal surfaces.
Study non-fillable curves in a hyperbolic surface with a real line.
problem Non-fillable curves in H2imesR method Asymptotic Plateau problem in H2imesR result First examples of non-fillable finite curves with no thin tail.
In this paper we investigate free boundary minimal surfaces in the unit ball in Euclidean 3-space, and by using holomorphic techniques we prove that intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Interpolates curves using maximal and minimal surfaces in different spaces.
problem Interpolating curves in spacelike and Euclidean spaces.
method Using maximal and minimal surfaces, interpolates curves based on the Björling problem.
result Constructs surfaces to interpolate curves in a specified manner.
Study on minimal submanifolds in curved spaces with unique solution to asymptotic Plateau problem.
problem Minimal submanifolds in negatively curved spaces with small curvature.
method Analysis of spheres at infinity and asymptotic Plateau problem.
result Complete minimal submanifolds bound a class of spheres with uniquely solvable asymptotic Plateau problem.
We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.
Minimal surfaces span periodic curves in 3D space.
problem Existence of minimal surfaces spanning periodic curves.
method Proof of existence for minimal surfaces using periodic curves in R3. result Existence of noncompact simply connected periodic minimal surfaces.
The paper proves Γ-convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves Γ-convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional. result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
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 …
Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.
Study delta invariant of minimal generic curves on rational surfaces.
problem Recover delta invariant of curve germs from surface singularity topology.
method Explicit formulae for minimal generic curves on rational surfaces, proving delta invariant values for quotient singularities.
result Explicit formulae and values for delta invariant of minimal generic curves on rational surfaces.
This study examines abnormal geodesics in 2D-Zermelo navigation problems, revealing their role in separating time minimal and maximal curves.
problem The role of abnormal geodesics in planar Zermelo navigation problems with strong current.
method Geometric time optimal control approach, focusing on the heading angle of the ship.
result Abnormal geodesics separate time minimal and maximal curves, and are both small-time minimizing and maximizing.
Study on abnormal curves in sub-Riemannian manifolds, proving length-minimizing properties.
problem Characterizing abnormal geodesics in sub-Riemannian manifolds.
method Analyzing curves that annihilate Lie brackets and proving minimization properties.
result Strictly abnormal geodesics can cease to be locally length-minimizing.
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.