Study on dual surfaces of rotational minimal and maximal in Euclidean and Lorentz-Minkowski spaces.
problem Investigating the duality between minimal and maximal surfaces in different spaces.
method Analysis of rotational surfaces and use of one-parameter group of rotations.
result Family of Bonnet minimal and maximal surfaces emerge in the duality process.
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
Study on unique minimal hypersurfaces in rotational domains.
problem Existence of compact free-boundary minimal hypersurfaces in rotational domains.
method Integral identity for compact free-boundary minimal hypersurfaces, applied to rotational domains.
result Existence of minimal hypersurfaces in rotational domains without topological restrictions.
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Minimal surfaces found in 4D space.
problem Minimal surfaces in 4D space.
method Reduced biharmonic equation to ODEs, excluded non-minimal solutions.
result Biharmonic rotational surfaces in 4D are minimal.
The paper characterizes special curves and generalizes rectifying-type curves in n-dimensional space.
problem Characterizing and generalizing special curves in higher dimensions.
method Characterization through Rotation minimizing frame (RMF) and generalization of rectifying-type curves.
result Rectifying-type curves are generalized in n-dimensional space.
The study examines singularities of ruled surfaces using RM vectors.
problem Analyzing singularities of ruled surfaces.
method Using Legendre curves and rotation minimizing frames.
result Classification of singularities of ruled surfaces.
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
The paper explores connections between three equations via Wick rotations and symmetries.
problem Investigating relations between solutions to specific equations under Wick rotations.
method Analyzing symmetries and transformations of solutions to the minimal surface, zero mean curvature, and Born-Infeld equations.
result Existence conditions and transformations of real and imaginary solutions under Wick rotations.
Explains minimal surfaces and their properties.
problem Understanding minimal surfaces and their characteristics.
method Analyzes various minimal surfaces and their properties.
result Discusses the properties and stability of minimal surfaces.
The paper confirms Yau's conjecture for minimal rotational hypersurfaces.
problem Yau's conjecture about the minimal area of certain hypersurfaces.
method Analyzes minimal rotational hypersurfaces to confirm the conjecture.
result The area of compact minimal rotational hypersurfaces is either equal to the unit sphere's area or another specific value.
We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…
Lie minimal surfaces are characterized by differential equations of principal curvatures.
problem Characterizing Lie minimal surfaces in Riemannian space forms.
method Using Euler-Lagrange equations and differential equations of principal curvatures.
result Rotational surfaces are found for certain relationships between principal curvatures.
General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…
New minimal surfaces in 4D space discovered using complex rotations.
problem Discovering new minimal surfaces in 4D space.
method Complex parabolic rotations of holomorphic null curves in 4C space.
result Existence of minimal surfaces foliated by conic sections in 4D space.
Study geometric properties of RM vector fields in Euclidean, Hyperbolic, and Kähler spaces.
problem Geometric properties of RM vector fields not fully explored.
method Analysis of RM vector fields along curves in Riemannian manifolds.
result Many geometric properties of RM vector fields are studied in specific Riemannian manifolds.
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.
Paper shows rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
problem Understanding rigidity of rotationally symmetric minimal hypersurfaces in 5D spaces.
method Pointwise hypersurface invariant analysis for minimal hypersurfaces in spaces of constant curvature.
result Rotationally symmetric minimal hypersurfaces in 5D spaces are rigid.
Study axisymmetric surfaces in Euclidean space for energy minimization.
problem Finding surfaces in Euclidean space that minimize energy.
method Phase plane analysis and maximum principle.
result Helicoidal stationary surfaces must be rotational.
The paper classifies CMC free boundary hypersurfaces in rotational domains.
problem Existence and uniqueness of free boundary constant mean curvature hypersurfaces in rotational domains.
method Classification and construction of CMC free boundary hypersurfaces under specific conditions.
result Classification of CMC free boundary hypersurfaces as topological disks or annuli.
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.
Study minimizes CR surfaces in Heisenberg group with rotational symmetry.
problem Minimizing CR surfaces with vanishing CR invariant energy E1 in Heisenberg group. method Proved local uniqueness, classified global surfaces with rotational symmetry, computed second variation.
result Clifford torus is not a local minimizer of E1. Characterizes spherical and plane curves using RM frames.
problem Understanding the geometric properties of curves in different spaces.
method Employing rotation minimizing frames to study curvature and torsion.
result Characterizes curves as those whose position vector lies on a moving plane.
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…
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
Upper bounds for Legendrian links in tight contact 3-manifolds.
problem Bounding the complexity of Legendrian links in tight contact 3-manifolds.
method Constructing exact Lagrangian cobordisms and defining minimal Lagrangian genus.
result Established upper bounds for Legendrian links with a common rotation number.
Study of minimal surfaces based on boundary geometry.
problem Understanding the shape of compact singular minimal surfaces from their boundaries.
method Estimates of area and height derived from boundary geometry; conditions for rotational surfaces; non-existence results for certain boundary configurations.
result Derived estimates and conditions for minimal surfaces based on boundary properties.
On a Riemannian 2-torus (T2,g) we study the geodesic flow in the case of low complexity described by zero topological entropy. We show that this assumption implies a nearly integrable behavior. In our previous paper \cite{GK} we already obtained that the asymptotic direction and therefore also the rotation number ex…
In this paper we finish the classification of rotational special Weingarten surfaces in S^2 x R and H^2 x R; i.e. rotational surfaces in S^2 x R and H^2 x R whose mean curvature h and extrinsic curvature K_e satisfy h=f(h^2-K_e), for some function f in C^1([0,+infty)) such that 4x(f'(x))^2<1 for any x>=0.
Proposes a new framework for learning image augmentations to improve classification performance.
problem Improving classification performance with a given class of predictors.
method Transformed Risk Minimization (TRM) framework that optimizes both predictive models and data transformations.
result Performance of TRM with SCALE algorithm compares favorably to prior methods on CIFAR10/100.
New minimal annuli found in unit ball, solving old problems.
problem Constructing free boundary minimal annuli in unit ball.
method Symmetric and foliated by spherical curvature lines.
result First non-embedded free boundary minimal annuli in unit ball.
New approach to rotational Weingarten surfaces using geometric momentum.
problem Classifying and characterizing rotational Weingarten surfaces.
method Introducing geometric linear momentum of a plane curve to reduce Weingarten conditions to differential equations.
result Classification of non-degenerate quadric surfaces and elasticoids.
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
We obtain isometric minimal helicoidal and rotational surfaces using generalized Bour's theorem in three dimensional Minkowski space. In addition, we show that the surfaces preserve minimality when their Gauss maps identically equal, choosing any diffentiable functions on the profile curve.
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
We show that a compact embedded minimal or constant mean curvature annulus with non-vanishing Gaussian curvature which is tangent to two spheres of same radius or tangent to a sphere and meeting a plane in constant contact angle is rotational.
A class of spiral minimal surfaces in E^3 is constructed using a symmetry reduction. The new surfaces are invariant with respect to the composition of rotation and dilatation. The solutions are obtained in closed form %through the Legendre transformation and their asymptotic behaviour is described.
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Constructs minimal annuli with free boundary in hyperbolic 3-space.
problem Finding minimal surfaces with boundary in hyperbolic geometry.
method Constructs families of non-rotational minimal annuli with shared symmetry.
result Bifurcates from hyperbolic catenoids, forming a countable collection.
Constructs minimal surfaces in a 3-ball using PDE gluing.
problem Finding minimal surfaces in a 3-ball with boundary constraints.
method PDE gluing construction of discrete free boundary minimal annuli.
result Discrete family of non-rotational free boundary minimal annuli in a unit 3-ball.
In this paper, we determine the maximally stable, rotationally invariant domains on the catenoids $\cC_a$ (minimal surfaces invariant by rotations) in the Heisenberg group with a left-invariant metric. We show that these catenoids have Morse index at least 3 and we bound the index from above in terms of the parameter $…
This paper connects Laguerre minimal surfaces to Weierstrass representations.
problem Understanding the relationship between Laguerre minimal surfaces and Weierstrass representations.
method Defining spherical mean curvature and providing Weierstrass-type representations for two classes of surfaces.
result Laguerre minimal surfaces are related to H2-surfaces, providing a new Weierstrass-type representation. We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show …
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand toru…
The paper computes spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
problem Computing spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
method Analyzing eigenvalues of second order Hill's equations and proving inequalities for stability index and eigenvalues.
result Proves that the stability index of minimal rotational examples is greater than 3n+4 and there are at least 2 positive Laplacian eigenvalues smaller than n. We study surfaces in Euclidean space R3 that are minimal for a log-linear density φ(x,y,z)=αx+βy+γy, where α,β,γ are real numbers not all zero. We prove that if a surface is φ-minimal foliated by circles in parallel planes, then these planes are orthogonal to the vector (α,β,γ) and the surface must…