The Minimax Sphere Eversion

by George Francis, John M. Sullivan Rob B. Kusner Chris Hartman & Glen Chappell

Published in Visualization and Mathematics, (H.-C. Hege, K. Polthier, eds.), Springer, 1997, pp 3-20 Plain-English Abstract

It was a surprising consequence of an abstract theorem of Smale that a spherical surface can be turned inside out without tearing or creasing, if we do allow the surface to pass through itself. Over the intervening forty years, mathematicians have designed different ways to see explicitly how such a ``sphere eversion'' can be achieved.

Here, we consider a new sphere eversion computed by minimizing an elastic bending energy for surfaces in space called the Willmore energy. We start with a complicated self-intersecting sphere (a Morin surface) which is halfway inside-out in the sense of having its inside and outside equally exposed. This halfway model is a saddle critical point for the Willmore energy. When we push off the saddle in the two directions and then flow downhill in energy to the ordinary round sphere, it is inside-out one way, but not the other way.

In any sphere eversion, the "double locus" of self-intersections is a key item to follow through time as the sphere turns inside out. Here, we can explicitly describe the double locus, and prove that our eversion is optimal not only geometrically, but also topologically in the sense of having the fewest catastrophes.