Half-way model

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.

A half-way model is an immersion of the sphere ${\displaystyle S^{2}}$ in ${\displaystyle \mathbb {R} ^{3}}$, 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.

Rob Kusner proposed optimal eversions using the Willmore energy on the space of all immersions of the sphere ${\displaystyle S^{2}}$ in ${\displaystyle \mathbf {R} ^{3}}$. 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).

...
Wikipedia