Minimax eversion


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 ; 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 in, 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 is the same process in reverse.

Explanation

proposed optimal eversions using the Willmore energy on the space of all immersions of the sphere in.
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.
Kusner's half-way models are saddle points for Willmore energy, arising 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. 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 :
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: .