If it looks like a sphere…Exploring the newly proposed solution to a famous problem about three-dimensional shapes

Science News, June 14, 2003 by Erica Klarreich

ERASING THE BAR If Thurston's conjecture can be proved, the Poincare conjecture will follow automatically. The logic goes more or less like this: In a closed three-dimensional space, if all loops of thread can be pulled tight to a point, mathematicians know that the only one of the eight geometries that can fit the space is spherical geometry. That means that no matter how convoluted the space appears, it must simply be a distorted version of the hypersphere.

After Thurston's work, mathematicians who wanted to prove the Poincare conjecture could focus on demonstrating that Thurston's vision of three-dimensional spaces is correct. By the early 1990s, Richard Hamilton of Columbia University had proposed a technique that he hoped would do just that--show that each three-dimensional space can be smoothed out into Thurston's special pieces. He defined a method, called the Ricci flow, for changing the shape gradually at each point to make the space more uniform. His equation resembles the physics equation that describes how heat spreads through a material.

"If you take a body where parts are hot and parts are cold and you let it stand, heat tends to flow by itself until the temperature is even," Milnor says. "In Hamilton's process, you have a manifold that is very curved in some places, maybe flat or negatively curved in other places, and you just let the curvature flow and try to even itself out."

For instance, the Ricci flow would make an egg-shaped surface gradually flatten out on the ends and bulge even more in the middle, getting closer and closer to a perfect sphere.

Hamilton was aware, however, that the flow would not always produce a uniform geometry. At any point in the space, the flow is determined mainly by the local geometry, not by the overall shape of the space. So, sometimes the geometry of one part of the space might change much faster than that of another part, producing a highly uneven geometry overall.

For example, picture a dumbbell--two weights connected by a thin bar--each portion of which is flowing with a mind of its own. The bar wants to even out its geometry with the weights to turn the whole thing into a nicely rounded sphere. Each weight, on the other hand, wants to make itself as spherical as possible. In the three-dimensional version of the dumbbell, depending on the initial geometry, the weights may predominate, growing rounder and rounder while the bar stretches into a long, thin neck.

Hamilton's idea for dealing with this difficulty was simply to snip out the neck at some appropriate point, continue the Ricci flow on the pieces, and glue the neck back in at the end. The resulting shape would have the right kinds of building blocks for Thurston's conjecture. But for more complicated shapes than the dumbbell, he couldn't show that these necks were the only extreme geometric forms the flow would produce. Other extremities, such as awkward protrusions he called cigars, might result.

What's more, perhaps every time the flow evened out one portion of the space, that portion's extreme shape would have moved somewhere else, like bulges in a rug that is being fit into a room too small for it. Extreme geometric features might cycle around and around, without the whole space ever growing uniform.