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Free boundary minimal Möbius bands in toroids
1Dipartimento di Matematica, Università di Trento, via Sommarive 14, Povo, 38123 Trento Italy.
Calculus of Variations and Partial Differential Equations
|June 26, 2026
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.
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.
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