This idea of rescaling the problem is what Uhlenbeck called Illustration from the Notices of the American Mathematical Society. You can find out more in Simon Donaldson's excellent article about Uhlenbeck's work.Ī schematic of bubbling: small regions around an awkward point are blown up into bubbles. Still possible to describe what happens there: rescaling the maps within small regions around the awkward points, youīlown up to what you might call "bubbles" in The sequence of minimising maps may not converge at some points on the sphere on which they are constructed, but there are at most finitely many such points and it's Uhlenbeck and Sacks showed that they do, at least mostly. Each of the slightly varied expressions comes with its own minimising map, so the idea is to see whether these minimising maps converge to anything meaningful regarding the original energy expression. You can construct a sequence of slightly-varied energy expressions that converges on the original energy expression. Means that for each of these slightly varied expressions you can find These new expressions can see scale and therefore regain the strength to prove a compactness The energy of the surface to obtain expressions that are ever so slightlyĭifferent (see the box). They tinkered with the usual expression for In a landmark paper, Uhlenbeck and Sacks found an ingenious The slightly varied energy expressions Uhlenbeck and Sacks were looking at have the form Where is the standard norm defined by the metrics on and and Given two compact manifolds and and a smooth map from to, the energy of the map is given by the formula Sphere to associate to which point on the surface, so there are many However, many different choices of which point on the Less-than-perfectly-spherical Earth by a map on a round globe. This is exactly what we do when we represent the Point on the round sphere is associated to a point on the deformed To find your way around such a wonky surface you might construct a map from a round sphere: every AsĪ loose illustration, imagine a surface that looks like a deformed sphere Ī knobbly potato, for example, or a deflated football. Problem lies with the many ways you can use to describe a surface. Strong enough to give you a similar result. When it comes to surfaces, however, energy isn't quite (called compact because there are limits) is important. It's not surprising that this kind of compactness result Minimal path from the infinity of all paths connecting A and B, Since the overarching problem is to wrestle a The concept of energy is intimately bound up with this convergence: if within a collection of smooth paths from A to B the energy of every path is smaller than some bound, then you can be sure that the collection contains a sequence of paths that converges to a continuous limit path from A to B. You can imagine infinitely many more blue paths getting closer and closer to the red one. Illustration of blue paths converging on a red path. Other, until they eventually converge on a limit path. Paths can bunch up, coming closer and closer to each When you are dealing just with paths the problem isĪ hugely helpful fact (in the setting Uhlenbeck and Sacks were considering) Is to find a surface which minimises this energy formula. Just as the length of a path between two points canīe described by an integral, so can the energy of a surface. One way of defining a minimal surface, which is easier to deal with than area, is in terms of somethingĬalled its energy (see the box below for a formal definition). Minimal surfaces, and I brought the main idea. "He brought the knowledge of the subject of "I didn't know very much about minimal surfaces, but we talkedĪbout them and worked together," Uhlenbeck said in a 2018 interview Uhlenbeck did her PhD in the calculus of variations in 1966 and turned herĪttention to minimal surfaces when she met Jonathan Sacks in theġ970s. The film taking all sorts of shapes, it could contain bumps forĮxample, but the flat shape it does take minimises area, and, equivalently, Within the frame after dipping has the smallest area possible. Plastic frame to be dipped into soapy water. The toys we use to blow soap bubbles come with a Stepping up a dimension, a similar problem lands you in the beautiful world Is all about minimising or maximising some quantity subject to constraints. You its length (for those who know calculus the formula is anīelongs to an area called the calculus of variations, which Given any path connecting the two points there's a formula giving Of us just take this fact for granted - how could it be otherwise? -īut it's also possible to prove it with full mathematical rigour. How could it be otherwise? Any other path is obviously longer! The shortest path between A and B runs along a straight line.
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