The Minimax Sphere Eversion

We consider an eversion of a sphere driven by a gradient flow for elastic bending energy. We start with a halfway model which is an unstable Willmore sphere with 4-fold orientation-reversing rotational symmetry. The regular homotopy is automatically generated by flowing down the gradient of the energy from the halfway model to a round sphere, using the Surface Evolver. This flow is not yet fully understood; however, our numerical simulations give evidence that the resulting eversion is isotopic to one of Morin’s classical sphere eversions. These simulations were presented as real-time interactive animations in the CAVE automatic virtual environment at Supercomputing’95, as part of an experiment in distributed, parallel computing and broad-band, asynchronous networking. 1 A History of Sphere Eversions To evert a sphere is to turn it inside out by means of a continuous deformation, a regular homotopy, which allows the surface to pass through itself, but forbids more serious singularities where the curvature becomes locally infinite. There have been many eversions since Smale [30] proved the possibility of this phenomenon forty years ago. The extraordinary difficulty in visualizing an explicit eversion has made it an effective challenge for a succession of both mathematical and graphical innovations. 2 Francis, Sullivan, Kusner, Brakke, Hartman, Chappell Our sphere eversion differs from all previous ones in that it proceeds automatically through an optimization procedure. We use the Surface Evolver, a computer program which is designed to solve variational problems, like finding the shape of soap films. We needed only to specify the starting point and a general strategy of energy minimization to the program. The resulting minimax eversion is optimal in that it requires the least bending at any stage. As in all good experiments, we did not know a priori that the evolver would be successful in producing a sphere eversion. To our great pleasure, it not only succeeded, but also produced an eversion which turned out to be among those designed by Bernard Morin three decades earlier [28]. That an optimal geometry for the eversion would match one envisioned by a pure topologist is an eloquent confirmation of mathematics. The sphere eversion story has been told many times from a variety of viewpoints. Chapter 6 of [9] discusses most of the work through 1986 primarily from the viewpoint of how to draw pictures of a process. For an update as well as a good review of the topological fundamentals, see Silvio Levy’s supplement [22] to the video Outside In [23]. Here we focus on computer animations, beginning with Nelson Max’s epic 16mm film [25]. It was generated as a frame-by-frame animation from a database of coordinates in R which had been entered manually [26]. The coordinates were taken from wire-mesh models made by C. Pugh, which depict, in eleven stages, what is to be perceived as a continuous surface moving through itself from a highly convoluted halfway model to a round sphere. This halfway model is an example of a Morin surface, an immersed sphere which has a 4-fold rotational symmetry that switches its orientation. (Thus if the two sides of this sphere were painted different colors, the quarter-turn would bring the surface back to its original position, but with the colors exchanged.) Morin recognized that a homotopy from a round sphere to such a surface automatically gives a sphere eversion; the second half of the eversion can be obtained from the first by reversing the temporal order, rotating the models 90◦ and switching colors. The resulting eversion has one kind of temporal symmetry; a fundamentally different kind figured in the first explicit sphere eversion (by Arnold Shapiro [12]) and the first illustrated publication of an eversion (by Tony Phillips [29]). These use the double cover of a Boy surface as a halfway model, and thus the symmetry exchanges only time and color. In 1977, Morin created an analytic parametrization [27] of his eversions. While it was conceptually correct, its implementation on a computer was visually unrecognizable. The first real-time, interactive computer animation was produced by John Hughes on a Stardent [19]. He fit spherical harmonics to polyhedral models of key stages in Morin’s eversion, obtaining a global parametrization by trigonometric polynomials. Morin and Denner developed a minimally polyhedral eversion [2], and Apéry [1] calculated harmoniously scaled algebraic formulas for its smooth companion. Both of these eversions were implemented as real-time interactive animations [11]. The Minimax Sphere Eversion 3 A truly new idea for an eversion, Bill Thurston’s corrugations [22], was expertly realized and stunningly rendered in the 1994 Geometry Center videotape, Outside In [23]. Unlike the earlier eversions, this method furnishes a general recipe for creating regular homotopies between surfaces whenever this is known to be theoretically possible. However, for all of these eversions, their designers had to know a priori what they should look like, and then search for formulas capturing this behavior. This limits the usefulness of their software tools in cases when one does not already know the whole story. We set about to build tools that were exploratory as well as explanatory. Our minimax eversion is generated automatically, with even the topological structure being chosen by the optimization procedure. 2 The Minimax Sphere Eversion 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 spot. 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 spot on a fence, but of course it is usually called a pass or saddle on roads through the mountains. It is precisely such a lowest-energy saddle that we encounter halfway through turning a sphere inside-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 the right side out to one inside-out. The energy which climbing kids and road engineers care about is the height they need to go above the surrounding territory. For 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. This energy is often called W , after Tom Willmore, who rekindled interest in W among mathematicians in the 1960s [31]. In 1983, Allen Hatcher [15] proved the Smale Conjecture, which in one formulation asserts that the diffeomorphism group of R is homotopy equivalent to the orthogonal group O(3). An equivalent formulation says that the space of embedded spheres in R is contractible. Interested in finding an analytic proof of this, Kusner started looking for functions with nice gradient flows on the configuration space of embedded (or immersed) surfaces. That W might be a good candidate follows from Robert Bryant’s [7] result: the only embedded W -critical sphere is round. The strategy for proving the Smale Conjecture analytically would be to flow an embedded sphere down the W -gradient until the sphere stopped changing, and thus, was round. Unfortunately, this flow does not preserve embeddedness. Bryant 1 A gradient flow for W would correspond to a fourth-order parabolic equation, which unlike a second-order equation, does not enjoy a maximum principle. 4 Francis, Sullivan, Kusner, Brakke, Hartman, Chappell had also found immersed W -critical spheres with self-intersections. Understanding these variationally led Kusner quite naturally to the idea of the minimax sphere eversion, because of some further important results already known about W . Willmore’s main result [31] was that W is uniquely minimized by round spheres, with the value 4π; any other surface M has energy W (M) greater than 4π. When the surface is immersed, there is a more general bound (due to Li and Yau [24], sharpened in [20]): If there is a point of R through which k “sheets” of a surface M pass, then W (M) is at least 4kπ; equality can occur here only if there is a complete minimal surface M̃ in R with k planar ends (“sheets at∞”), and a Möbius transformation which carries M̃ to M (and ∞ to the k-fold point on M). Taking k = 1 and M̃ to be a flat plane, this implies Willmore’s original result. The proof makes use of the fact that the quantity W + 4kπ (where k is the multiplicity of the surface at ∞) is invariant under Möbius transformations of R ∪∞. Bryant’s classification [7] of W -critical spheres showed that the lowest saddle point for W is at 16π, realized by a certain family of immersed spheres with one quadruple point. By the above, each such surface arises as the Möbius transformation of some minimal surface in R with four planar ends. In the meantime, Tom Banchoff and Nelson Max had shown [3] (see also [18]) that every sphere eversion must pass through an immersed sphere with at least one quadruple point. Thus, by the Li-Yau inequality above, every sphere eversion must pass over the W = 16π level. And if some W critical sphere at this 16π level were a saddle point (rather than a local minimum), then we could simply flow to either side of the saddle (in the most negative Hessian direction) by a W -gradient flow, and the flow would have to proceed down to the W -minimizing round sphere on either side. Note that if the saddle surface halfway model has an orientation-reversing symmetry, then these two round spheres must have the opposite orientation. So, by climbing 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. In 1977 Francis circulated an illustrated manuscript, see [9, Ch. 6], about sphere eversions equivariant under all the cyclic rotation groups, the tobacco pouch eversions. This inspired Kusner to find an infinite family of W -critical spheres (with even-order cyclic symmetry) and real projective planes (with odd-order symmetry). These are Möbius images of complete minimal surfaces with planar ends for which there is explicit, symmetric Weierstrass data [20]. They include a W -minimizing Boy surface with 3-fold symmetry, as well as the Morin surface of 4-fold symmetry that we use as our halfway model h0. 2 Bryant’s proof can be simplified and unified using the spinor representation and an abstract skew-form [21]. The Minimax Sphere Eversion 5 The image M of this immersion h0 is a Möbius transformation of a minimal surface M̃ whose Weierstrass representation can be integrated explicitly. The surface M̃ is given as an immersion h̃ of the Riemann sphere S with four punctures into R: h̃(w) = Re (( i(w − w), (w + w), i 2 (w 4 + 1) )