Kids climbing fences, along with engineers building mountain roads and scientists rocketing to the moon, know that the easiest way to get from one side to the other (and back again) is to follow the path which goes over the lowest place. This is so obvious to kids that they don't have a name for - or at least they don't tell their parents about - this place on a fence, but of course this place is usually called a ``pass'' on roads through the mountains. It is precisely such a lowest energy pass that we encounter halfway through turning a sphere in-side-out via the ``minimax sphere eversion.'' Indeed, the minimax sphere eversion might be viewed as the ``easiest'' path of immersed spheres leading from a round sphere with out-side-out to one with out-side-in. The energy which climbing kids and road engineers care about is the height they need to go above the surrounding territory. For us mathematicians interested in everting spheres, another energy is needed: the ``elastic bending energy,'' which assigns to any immersed surface the integral of the square of the mean curvature. For historical reasons this energy is often called W, after Tom Willmore, who rekindled interest in W among mathematicians.
I began working with the elastic bending energy W in the early 1980's when I was a graduate student at Berkeley learning low dimensional topology from Rob Kirby, and starting to work on minimal surfaces and variational problems with Rick Schoen. I was looking for functions with nice gradient flows on the configuration spaces of embedded or immersed surfaces. This was motivated in part by Allen Hatcher's (then recent) proof of the Smale Conjecture, which in one formulation asserts that the diffeomorphism group of R^3 is homotopy equivalent to the orthogonal group O(3). An equivalent form of the Smale Conjecture is:
The space of embedded spheres in R^3 is contractible.
I was interested in giving an analytic proof of this with some kind of gradient flow for W, using a key fact which had just been proven by Robert Bryant:
The only embedded W-critical sphere is round.
My strategy was to start with any embedded sphere and run some (negative) W-gradient flow till the sphere stopped flowing, and thus, was round. The main problem I ran into was that any reasonable W-gradient flow need not preserve embeddedness - this is because the flow corresponds to a fourth-order parabolic equation (second-order parabolic equations enjoy a maximum principle which maintains embededness) - and Bryant had found immersed W-critical spheres with self intersections. Since other, more subtle issues (such as perturbing W slightly to ensure certain compactness properties of the flow) also were needed, I set aside this approach to the Hatcher Theorem.
Nevertheless, there were other nice results of Bryant about immersed W-critical spheres which I wanted to understand variationally. This leads fairly directly to the idea of the minimax sphere eversion.
To begin, let me review a couple of nice properties of W, the simplest of which is:
W is uniquely minimized by round spheres, with the value 4\pi; any other surface S has energy W(S) greater than 4\pi.
Another property that W enjoys is an inequality (discovered by Li and Yau, and sharpened by me) that can be used to control the complexity of the immersed surface:
If there is a point of R^3 through which k ``sheets'' of S pass, then W(S) is at least 4k\pi; the only way equality can occur here is if there is a complete minimal surface S' in R^3 with k planar ends - `` sheets at infinity'' - and a Mobius transformation which carries S' to S (and infinity to the k-uple point on S).
Notice that the first property follows from the second one, where k=1 and S' is a flat plane. The proof makes use of the fact that the quantity W + 4k\pi (where k is the multiplicity of the surface at infinity) is invariant under Mobius transformations of R^3.
The interesting result Bryant had shown was that for immersed spheres, the lowest critical value for W is 16\pi, realized by a certain family of W-critical spheres with one quadruple point. By the above, each surface arises from Mobius inversion of some minimal surface in R^3 with 4 planar ends. (Part of Bryant's explicit algebra can be simplified and unified using an abstract skew-form invented by Nick Schmitt, or even replaced by a clever topological argument - see my recent paper with Schmitt on the Spinor Representation of Minimal Surfaces for arguments of each kind.)
Now I had also become aware around 1982, through conversations with John Hughes, who was finishing up his Berkeley thesis, about this nice fact, due to Tom Banchoff and Nelson Max:
Every sphere eversion must pass through an immersed sphere with at least one quadruple point.
Thus, every sphere eversion must pass over the magic W=16\pi level, by the Li-Yau inequality above. And (more magic) if some W-critical sphere at this 16\pi level were a saddle point, then we could simply flow to either side of the saddle (in the most negative Heesian direction) by a W-gradient flow, and the flow would have to proceed down to the W-minizing round sphere on either side - and these two round spheres would have the opposite orientation. So, by climbing back up the (positive) W-gradient flow, over the saddle and back down the other side, one would get an ``optimal'' sphere eversion - the minimax sphere eversion!
End of story? Not quite.
Hughes shared with me a beautifully illustrated manuscript by George Francis about sphere eversions equivariant under all the cyclic groups, and this inspired me to find an infinite family of W-critical spheres (even order) and real projective planes (odd order), and their corresponding complete minimal surfaces with planar ends - replete with symmetric Weierstrass data - including my W-minimizing Boy's surface with 3-fold symmetry, that I wrote about in my 1987 AMS Bulletin article on Conformal Geometry and Complete Minimal Surfaces, and which Michael Callahan, Jim Hoffman and I made some pictures of at ``stone-age'' GANG in 1986.
But these ``still'' images of eversion midpoints were not very satisfying to me - I wanted to get people to animate this minimax eversion, but in the mid 1980's nobody that I knew had developed effective software for modelling this kind of gradient flow. And I had other math to work on, so the idea ``sat on a shelf'' until....
In 1989 I was asked to help organize the Five Colleges Regional Geometry Institute, and in particular to direct the first summer (1991) of the research program on the topic of computation in geometry and geometric variational problems. Ken Brakke and his Evolver were star attractions, and under the prodding of several of us, including Lucas Hsu, Ivan Sterling, John Sullivan and myself, Brakke kindly agreed to let surfaces evolve according to motions other than area-gradient flow! Indeed, that summer Brakke worked out, and programmed into Evolver, the formulas for discretized W-gradient flow; Brakke's Evovler groupies created Evolver datafiles and evolved them (almost forever!) there at Five Colleges; and later, at GANG and at the Geometry Center. A subset of us (Hsu, Sullivan and I) eventually wrote up some of our experiments minimzing W on surfaces of higher genus in Volume 1 of David Epstein's new Journal of Experimental Mathematics in 1992. And to animate some of these evolutions for ``general audiences'', Jim Hoffman and I made the video Elastic Surfaces and Conformal Geometry, shown at Berkeley in October 1992, which led to the fateful conversation there at MSRI among Francis and Sullivan and me, where we decided to do it! - to animate the minimax sphere eversion!!
That took only 3 more years, and relied heavily upon Brakke's development of the Evolver Hessian method, which finds that negative eigendirection needed to push the eversion midpoint saddle surface - computed from my inverted Weierstass data - to each side; to Sullivan's expertise in harnessing the ``beast'' and riding the saddle down to the minimum of W; to all of Francis' panache and organizational skills; and to my [who knows what?]; to yield the ultimate form you see here...