Min-Max Theory of Minimal Surfaces and Phase Transitions

The min–max theory of minimal surfaces is a variational framework that produces critical submanifolds of minimal area by optimising over families—or “sweepouts”—of candidate surfaces. Originating in geometric measure theory, it has been refined through the Almgren–Pitts programme and subsequent analytic advancements to furnish existence and index estimates for closed or free‐boundary minimal hypersurfaces in Riemannian manifolds. In parallel, phase transition models—most notably the Allen–Cahn functional—have emerged as diffuse approximations of the area functional, with solutions forming “diffused interfaces” whose vanishing‐limit measures concentrate on minimal hypersurfaces. The convergence of these two perspectives has yielded profound insights: phase–field solutions obtained via one‐parameter min–max constructions can be shown to converge, under energy and index bounds, to multiplicity‐one minimal hypersurfaces. Meanwhile, the introduction of p-widths furnishes a nonlinear spectral invariant, akin to eigenvalues, that organises the breadth of minimal surface geometry in a given manifold. Together, these approaches interlink variational topology, nonlinear elliptic theory and geometric measure theory, advancing our understanding of the existence, stability and Morse index of minimal surfaces, as well as guiding practical computations of minimal shapes in applied contexts ranging from materials science to general relativity.

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Min-Max Theory of Minimal Surfaces and Phase Transitions publication trend

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Technical terms

Min–max theory: A variational method that finds critical points of a functional by optimising over continuous families of candidate objects.

Minimal surface: A surface or hypersurface whose mean curvature vanishes identically, corresponding to a stationary point of the area functional.

Allen–Cahn functional: A phase transition energy whose diffuse‐interface solutions approximate minimal hypersurfaces in the sharp‐interface limit.

p-width: A nonlinear invariant of a Riemannian manifold defined via the min–max of the area functional over p-parameter sweepouts.

Varifold: A generalised surface in geometric measure theory represented by a measure on the space of tangent planes, allowing treatment of singular minimal sets.

Sweepout: A continuous family of surfaces parametrised to “sweep” through a manifold, used to define min–max values of the area functional.

References

1. The p-widths of a surface. Publications mathématiques de l'IHÉS (2023).
2. Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature. Communications on Pure and Applied Mathematics (2023).
3. Plateau’s problem via the Allen–Cahn functional. Calculus of Variations and Partial Differential Equations (2024).