Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Magnetostatic Boundary Conditions01:28

Magnetostatic Boundary Conditions

An electric field suffers a discontinuity at a surface charge. Similarly, a magnetic field is discontinuous at a surface current. The perpendicular component of a magnetic field is continuous across the interface of two magnetic mediums. In contrast, its parallel component, perpendicular to the current, is discontinuous by the amount equal to the product of the vacuum permeability and the surface current. Like the scalar potential in electrostatics, the vector potential is also continuous...
Toroids01:27

Toroids

A toroid is a closely wound donut-shaped coil constructed using a single conducting wire. In general, it is assumed that a toriod consists of multiple circular loops perpendicular to its axis.
When connected to a supply, the magnetic field generated in the toroid has field lines circular and concentric to its axis. Conventionally, the direction of this magnetic field is expressed using the right-hand rule. If the fingers of the right hand curl in the current direction, the thumb points in the...
Torque On A Current Loop In A Magnetic Field01:13

Torque On A Current Loop In A Magnetic Field

The most common application of magnetic force on current-carrying wires is in electric motors. These consist of loops of wire, which are placed between the magnets with a magnetic field. When current flows through the loops, the magnetic field applies torque, which causes the shaft to rotate, thus converting electrical energy to mechanical energy.
Consider a rectangular current-carrying loop containing N turns of wire, placed in a uniform magnetic field. The net force on a current-carrying loop...
Boundary Conditions for Current Density01:25

Boundary Conditions for Current Density

Current density becomes discontinuous across an interface of materials with different electrical conductivities. The normal component of the current density is continuous across the boundary.
Mohr's Circle for Plane Strain01:18

Mohr's Circle for Plane Strain

Mohr's circle is a crucial graphical method used to analyze plane strain by plotting strain on a set of cartesian coordinates, where the abscissa is normal strain ∈ and the ordinate is shear strain γ. Similarly to Mohr’s circle for plane stress, two points X and Y are plotted. Their coordinates are (∈x, -γXY) and (∈Y, γXY), respectively.
Mohr's circle visually represents the strain states under various conditions, which is essential for understanding material behavior. The center of Mohr's...
Boundary Conditions: Lossless Lines01:21

Boundary Conditions: Lossless Lines

Consider a single-phase, two-wire, lossless transmission line terminated by an impedance at the receiving end and a source with Thevenin voltage and impedance at the sending end. The line, with length, has a surge impedance and wave velocity determined by the line's inductance and capacitance.
At the receiving end, the boundary condition states that the voltage equals the product of the receiving-end impedance and current. This relationship is expressed as a function of the incident and...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Existence of nontrival <i>n</i>-harmonic maps via min-max methods.

Calculus of variations and partial differential equations·2026
Same journal

On De Giorgi's conjecture of nonlocal approximations for free-discontinuity problems: The symmetric gradient case.

Calculus of variations and partial differential equations·2026
Same journal

Thermo-elastodynamics of nonlinearly viscous solids.

Calculus of variations and partial differential equations·2026
Same journal

JKO schemes with general transport costs.

Calculus of variations and partial differential equations·2026
Same journal

Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling.

Calculus of variations and partial differential equations·2026
Same journal

Energy identity and no neck property for <math><mi>ε</mi></math> -harmonic and <math><mi>α</mi></math> -harmonic maps into homogeneous target manifolds.

Calculus of variations and partial differential equations·2026

Related Experiment Video

Updated: Jun 27, 2026

Optimized Fabrication Procedure for High-Quality Graphene-based Moir&#233; Superlattice Devices
11:24

Optimized Fabrication Procedure for High-Quality Graphene-based Moiré Superlattice Devices

Published on: July 11, 2025

Free boundary minimal Möbius bands in toroids.

Mario B Schulz1

  • 1Dipartimento di Matematica, Università di Trento, via Sommarive 14, Povo, 38123 Trento Italy.

Calculus of Variations and Partial Differential Equations
|June 26, 2026
PubMed
Summary

This study proves round, mean convex toroids contain infinite free boundary minimal Möbius bands and annuli. These surfaces, found using equivariant variational methods, have areas scaling with symmetry group order.

Area of Science:

  • Differential Geometry
  • Geometric Analysis
  • Topology

Background:

  • Minimal surfaces are central to geometric analysis.
  • Toroids of revolution are a key class of surfaces studied in geometry.
  • Free boundary minimal surfaces present significant mathematical challenges.

Purpose of the Study:

  • To investigate the existence of free boundary minimal Möbius bands and annuli on toroids.
  • To explore the relationship between surface area and symmetry in minimal surface theory.
  • To utilize advanced variational methods for constructing novel minimal surfaces.

Main Methods:

  • Application of equivariant variational methods.
  • Analysis of free boundary conditions on minimal surfaces.
  • Exploitation of symmetry properties of toroids of revolution.
Keywords:
49Q2053A1058E12

More Related Videos

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

Related Experiment Videos

Last Updated: Jun 27, 2026

Optimized Fabrication Procedure for High-Quality Graphene-based Moir&#233; Superlattice Devices
11:24

Optimized Fabrication Procedure for High-Quality Graphene-based Moiré Superlattice Devices

Published on: July 11, 2025

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations
13:56

Probe Type II Band Alignment in One-Dimensional Van Der Waals Heterostructures Using First-Principles Calculations

Published on: October 12, 2019

Main Results:

  • Proof of infinitely many geometrically distinct embedded free boundary minimal Möbius bands on toroids.
  • Demonstration of infinitely many embedded free boundary minimal annuli on toroids.
  • Established linear growth of surface areas with the order of symmetry groups.

Conclusions:

  • Round, strictly mean convex toroids of revolution harbor rich families of minimal surfaces.
  • Equivariant variational methods are effective for constructing complex minimal surfaces.
  • The study advances understanding of minimal surface topology and area properties.