*by Rob Kusner*

(This is a lightly edited version of
my article
in the
2007 UMass Math Department Newsletter
with links to references
to be included; thanks to Paul Gunnells for his helpful comments and for the
figure.)

Imagine a clear sky on a moonless night in a place like Amherst. Being
mathematicians, we ignore the few thousand nearby stars and the faint
glow of our Milky Way or other nearby galaxies. What remains is the
uniformly dark background, which any of us can observe directly. This
basic observation supports a cosmic hypothesis: on a large scale, we
live in a space which looks the same in all directions and the same at
all points and which is locally like the up-down, front-back,
left-right world of everyday experience. In more mathematical terms,
our present universe can be modeled by an isotropic and homogeneous
three-manifold.

What are the consequences of this mathematical model of the cosmos?
The first is that such a manifold has constant curvature. This means
that the laws of trigonometry are the familiar ones from Euclidean
(flat, or zero-curvature) geometry, or their spherical
(positive-curvature) or hyperbolic (negative-curvature)
counterparts. We could even measure this curvature
experimentally. Indeed, if we are only interested in the sign of the
curvature, the experiment is very simple: construct a triangle in
space with straight (geodesic) edges and add up the vertex angles;
when the angle sum is exactly 180 degrees, space is flat; otherwise,
space is positively or negatively curved when this angle sum is
greater or less than 180 degrees, respectively. The great German
mathematician C. F. Gauss carried out this experiment and found,
within the precision of his early 19th century instruments, that space
appeared to be flat. This is a rather famous result: both Gauss and
his experiment were depicted on the last series of 10-mark notes in
Germany before the euro became the currency there.

By the turn of the 20th century, mathematicians like H. Poincare,
F. Klein, and D. Hilbert had become interested in classifying these
three-manifolds, both as cosmological models and for their independent
mathematical interest. Of particular importance are the closed
three-manifolds of constant curvature; these manifolds have no
boundary but are of finite extent. The simplest example is the round
three-sphere, which has positive curvature. This can be thought of as
the unit-length vectors in four-dimensional space, but another useful
way to visualize the three-sphere is as a three-ball with all of its
boundary identified to a single point.

Another example of a closed three-manifold of constant curvature is
the three-torus. This is a flat three-manifold obtained by gluing the
top and bottom, front and back, and left and right faces of a cube,
like a cosmic version of the Mario Brothers video game; this space
parametrizes the configurations of a carpenter's folding ruler with
four segments and three pivots (the typical ruler with a dozen
segments and eleven pivots corresponds to an eleven-torus)! Even if
you've never played with a carpenter's ruler you can imagine something
of what life would be like if the universe were shaped like a
three-torus. If you were to stand in a three-torus and shine a
flashlight straight ahead, instead of disappearing into darkness the
light would illuminate the back of your head! This is because the
light leaves the front face of the cube and re-enters the back face,
which was glued to the front to form the three-torus. Similarly
shining the light down at your feet would reveal the top of your head,
and shining the light to the right would light up your left side
(hopefully your good side). The effect is something like being in a
completely mirrored room, except that instead of your reflected image
you see a parallel translation of yourself. The three-torus is a
fairly complicated three-manifold, but over the last hundred years
topologists busily manufactured even more exotic examples. For
instance, instead of a cube, take a solid dodecahedron. This is the
famous geometric object from antiquity with 12 pentagonal faces that
forms the basic structure underlying a soccer ball. Glue the opposite
faces of a dodecahedron together in pairs by twisting each 18 degrees
clockwise (you might need a rather stretchy dodecahedron for this).
This yields a three-manifold of positive curvature originally
discovered by Poincare and today called the Poincare dodecahedral
space in his honor.
This figure shows something of what this
space looks like, although it's hard to draw in our mundane three-
dimensional world. The dodecahedral space has the remarkable property
that every loop bounds a surface, although actually proving this
requires some delicate algebra. Indeed, at first Poincare thought
this space was a counterexample to his conjecture.

Around the same time, and with some friendly competition from Poincare
and Hilbert, A. Einstein had the revolutionary idea that space is not
static but dynamic, and that its geometry or metric evolves by a wave
equation that now bears his name. Although the Einstein equation in
general remains difficult to understand mathematically, special cases
have been solved when the space-like slices are three-manifolds of
constant curvature. For example, the deSitter model, where all the
slices are three-spheres of time-varying constant curvature, is
currently employed to explain the observed Hubble expansion of the
universe.

As typical of the long, give-and-take tradition between physics and
geometry, mathematicians began to use this new physical intuition to
conjecture and prove results about the geometry and topology of
manifolds with natural curvature-conditions. For example, in
Einstein's theory, positive energy-density translates into a
positivity condition on curvature, and the Ricci curvature in
particular. This led to one thread of research, starting in the 1930s,
when S. Bochner and A. Lichnerowicz showed that Ricci curvature is
closely related to the topology of a closed manifold (positive Ricci
curvature implies the vanishing of the first homology, for
example). Around 1940 S. Myers proved that positive Ricci curvature
implies finite fundamental group (the more refined algebraic structure
invented by Poincare when he realized that homology alone could not
distinguish the three-sphere from the dodecahedral-space). In fact,
Myers observed that Ricci curvature is enough to control the average
focusing of geodesics, an argument that inspired the much more recent
singularity theorems of R. Penrose and S. Hawking in general
relativity. By the 1970s my advisor R. Schoen, along with L. Simon,
S. T. Yau, and my colleague and collaborator Bill Meeks had taken up
minimal surface methods (instead of using geodesics) to generalize
many of these ideas to manifolds of positive scalar curvature. And
perhaps the culmination of this thread of work, with important
feedback into physics, has been the proofs of the Positive Mass
Theorem and more recently the Penrose Inequality by Schoen, Yau, and
their students.

Another thread leads rather directly from Einstein's ideas of
cosmological evolution to the geometric ideas that were developed by
R. Hamilton and G. Perelman to solve the Poincare and Geometrization
Conjectures. To get a flavor of this, we return to the period just
before 1920 when Hilbert had realized that the equations of general
relativity were of a variational nature: criticality for the total
scalar curvature functional (perhaps with a volume constraint) gives
rise to the Einstein equations for the geometry. Thus a natural way to
get a nice geometry on a three-manifold might be to use the direct
method in the calculus of variations to find such critical
metrics. This approach was attempted by H. Yamabe in 1950s, but it is
quite subtle since the critical metrics are infinitely unstable: along
conformal changes of metric one can minimize the total scalar
curvature functional (this became known as the Yamabe problem, solved
by Schoen in the early 1980s), but in transverse directions one must
maximize this functional to achieve an Einstein metric, which on a
three-manifold is a metric of constant curvature. This maximization
problem has not yet been solved, although M. Anderson obtained some
partial results in that direction about a decade ago.

What was Hamilton's successful approach? Instead of focusing on the
critical metric itself, he studied the dynamical flow that would
evolve to such a critical metric. And his key insight, based on years
of experience with the heat flow for harmonic maps, was to replace the
Einstein wave equation with an analogous heat equation for the metric,
known as the Ricci flow, which treats the curvature like a molten material
to be equitably redistributed around the manifold.
Hamilton's first result in the early 1980s
recovered the theorem of Myers as a special case, at least in
dimension three: when the Ricci curvature is positive, the Ricci flow
evolves a metric on a three-manifold to one of constant positive
curvature, like that on the three-sphere or the dodecahedral space.

Later in the 1980s, Hamilton, spurred on by Yau, began in earnest to
apply his Ricci-low methods to study the three-dimensional Poincare
Conjecture as well as the more general Geometrization Conjecture for
three-manifolds made in the 1970s by W. Thurston. This conjecture is a
natural outgrowth of the ideas from the time of Poincare, Klein,
Hilbert, and Einstein, asserting that any three-manifold can be
decomposed into pieces that carry either a constant-curvature geometry
(recall that means homogeneous and isotropic), or else a slightly less
symmetric geometry that is merely homogeneous. There are five
additional types of the latter geometry. Although explaining all this
in detail would, to borrow a line from Poincare, "take us too far
away," in effect what Hamilton began to realize (and what Perelman
perfected) was that while the constant curvature geometries arise as
steady states of the Ricci flow, the other five homogeneous geometries
arise naturally where the dynamics of the Ricci flow is more
complicated and where topological changes (neck pinching or surgery:
physicists might call these "wormholes") happen. This picture is not
yet completely clear, even to the experts, so this is probably a good
place to stop....

**Address**

1435G Lederle Graduate Research Tower Department of Mathematics University of Massachusetts at Amherst Amherst MA 01003 USA

**Phone** 413 545 6022

**Lab** 413 545 4605

**Secretary** 413 545 0510

**Fax** 413 545 1801

**Electronic Mail** rob@gang.umass.edu

All material on this website is Copyleft* by Rob Kusner.

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