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| In [[geometry]], '''minimax eversions''' are a class of [[sphere eversion]]s, constructed by using half-way models.
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| It is a [[calculus of variations|variational]] method, and consists of special homotopies (they are shortest paths with respect to [[Willmore energy]]); contrast with Thurston's corrugations, which are generic.
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| 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.
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| Eversions via half-way models are called ''tobacco-pouch eversions'' by Francis and Morin.
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| ==Half-way models==
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| A half-way model is an immersion of the sphere <math>S^2</math> in <math>\R^3</math>, 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.
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| ==Explanation==
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| [[Rob Kusner]] proposed optimal eversions using the [[Willmore energy]] on the space of all [[immersion (mathematics)|immersions]] of the sphere <math>S^2</math> in <math>\mathbf{R}^3</math>.
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| 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).
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| Kusner's half-way models are [[saddle point]]s 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.
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| There are two families of half-way models (this observation is due to Francis and Morin):
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| * 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).
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| * 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
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| ==History==
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| 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: [http://torus.math.uiuc.edu/jms/Papers/isama/color/opt3.htm historical details].
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| ==References==
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| * [http://torus.math.uiuc.edu/jms/Papers/isama/color/opt3.htm Bending Energy and the Minimax Eversions] (in John M. Sullivan's [http://torus.math.uiuc.edu/jms/Papers/isama/color/ "The Optiverse" and Other Sphere Eversions])
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| [[Category:Differential topology]]
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