Minimax eversion
title: "Minimax eversion" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["differential-topology"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Minimax_eversion" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models.
It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston's corrugations, which are generic.
The original method of half-way models was not optimal: the regular homotopies passed through the midway models, but the path from the round sphere to the midway model was constructed by hand, and was not gradient ascent/descent.
Eversions via half-way models are called tobacco-pouch eversions by Francis and Morin.
Half-way models
A half-way model is an immersion of the sphere S^2 in \R^3, which is so-called because it is the half-way point of a sphere eversion. This class of eversions has time symmetry: the first half of the regular homotopy goes from the standard round sphere to the half-way model, and the second half (which goes from the half-way model to the inside-out sphere) is the same process in reverse.
Explanation
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Rob Kusner proposed optimal eversions using the Willmore energy on the space of all immersions of the sphere S^2 in \mathbf{R}^3. The round sphere and the inside-out round sphere are the unique global minima for Willmore energy, and a minimax eversion is a path connecting these by passing over a saddle point (like traveling between two valleys via a mountain pass).
Kusner's half-way models are saddle points for Willmore energy, arising (according to a theorem of Bryant) from certain complete minimal surfaces in 3-space; the minimax eversions consist of gradient ascent from the round sphere to the half-way model, then gradient descent down (gradient descent for Willmore energy is called Willmore flow). More symmetrically, start at the half-way model; push in one direction and follow Willmore flow down to a round sphere; push in the opposite direction and follow Willmore flow down to the inside-out round sphere.
There are two families of half-way models (this observation is due to Francis and Morin):
- odd order: generalizing Boy's surface: 3-fold, 5-fold, etc., symmetry; half-way model is a double-covered projective plane (generically 2-1 immersed sphere).
- even order: generalizing Morin surface: 2-fold, 4-fold, etc., symmetry; half-way model is a generically 1-1 immersed sphere, and a twist by half a symmetry interchanges sheets of the sphere
History
The first explicit sphere eversion was by Shapiro and Phillips in the early 1960s, using Boy's surface as a half-way model. Later Morin discovered the Morin surface and used it to construct other sphere eversions. Kusner conceived the minimax eversions in the early 1980s: historical details.
References
- Bending Energy and the Minimax Eversions (in John M. Sullivan's "The Optiverse" and Other Sphere Eversions)
References
- J. Scott Carter. (2012). "An Excursion in Diagrammatic Algebra: Turning a Sphere from Red to Blue". World Scientific.
- Michele Emmer. (2005). "The Visual Mind II". MIT Press.
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