Geometric Measure Theory and Plateau Problems

Geometric measure theory provides a rigorous framework to study surfaces and their generalisations using measure‐theoretic tools. At its core lies the notion of mass minimisation: finding objects of least “area” within a prescribed homological or topological class. Plateau’s problem, in its classical form, asks for a surface of minimal area spanning a given boundary curve. In modern terms this becomes a search for an integral current or varifold that minimises its mass subject to a boundary constraint. Existence is typically secured by compactness arguments and lower semicontinuity of the mass functional, while regularity theory addresses smoothness except on a singular set of controlled dimension. Key advances have extended classical results to higher codimension, arbitrary multiplicity and anisotropic surface energies, and have explored connections with partial differential equations, shape optimisation and material science. Across these developments, the interplay of measure‐theoretic compactness, geometric variational techniques and local regularity estimates underpins a coherent picture of how minimal and almost‐minimal structures behave both in the interior and at their boundaries.

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Geometric Measure Theory and Plateau Problems publication trend

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Technical terms

Integral current: A generalised oriented surface with integer multiplicity, represented as a functional on differential forms, used to model minimal submanifolds in measure‐theoretic terms.

Varifold: A measure on the space of k‐planes in Euclidean space, encoding unoriented surface-like objects and their first variation without integer weights.

Mass minimisation: The variational principle of seeking a current or varifold that minimises the total “area” (mass) subject to prescribed boundary or homology constraints.

Mean curvature: The first variation of area per unit normal displacement; zero mean curvature characterises a minimal surface in the classical sense.

Multiplicity: The integer weight assigned to a current at a given point, representing how many sheets overlap in a mass‐minimising configuration.

References

1. An Allard-type boundary regularity theorem for 2d minimizing currents at smooth curves with arbitrary multiplicity. Publications mathématiques de l'IHÉS (2024).
2. Geometric measure theory and differential inclusions. Annales de la faculté des sciences de Toulouse Mathématiques (2021).
3. On the constancy theorem for anisotropic energies through differential inclusions. Calculus of Variations and Partial Differential Equations (2021).
4. Soap films with gravity and almost-minimal surfaces. Discrete and Continuous Dynamical Systems (2019).