The Poincaré Conjecture Clay Research Conference Resolution of the Poincaré Conjecture Institut Henri Poincaré Paris, France, June 8–9, 2010 - bet 20
Is there a (quasi)-canonical way of associating statistical ensembles
S to geometric system S of PDE, such that the equations emerge at low temperatures
T and also can be read from the properties of high temperature states of
S by some
“analytic continuation” in T ?
The architectures of liposomes and micelles in an ambient space, say W , which
are composed of “somethings” normal to their surfaces X
⊂ W , are reminiscent of
Thom-Atiyah representation of submanifolds with their normal bundles by generic
∶ W → V
, where V
is the Thom space of a vector bundle V
and where manifolds X
) ⊂ W come with their normal bundles
induced from the bundle V
The space of these “generic maps” f
looks as an intermediate between an
individual “deterministic” liposome X and its high temperature randomization.
Can one make this precise?
e-Sturtevant Functors. All that the brain knows about the geom-
etry of the space is a ﬂow S
of electric impulses delivered to it by our sensory
organs. All what an alien browsing through our mathematical manuscripts would
directly perceive, is a ﬂow of symbols on the paper, say G
Is there a natural functorial-like transformation
P from sensory inputs to
mathematical outputs, a map between “spaces of ﬂows”
P ∶ S → G such that
It is not even easy to properly state this problem as we neither know what our
“spaces of ﬂows” are, nor what the meaning of the equality “=” is.
Yet, it is an essentially mathematical problem a solution of which (in a weaker
form) is indicated by Poincar´
e in . Besides, we all witness the solution of this
problem by our brains.
An easier problem of this kind presents itself in the classical genetics.
What can be concluded about the geometry of a genome of an
organism by observing the phenotypes of various representatives
of the same species (with no molecular biology available)?
This problem was solved in 1913, long before the advent of the molecular biology
and discovery of DNA, by 19-year old Alfred Sturtevant (then a student in T.
H. Morgan’s lab) who reconstructed the linear structure on the set of genes on a
chromosome of Drosophila melanogaster from samples of a probability measure on
the space of gene linkages.
Here mathematics is more apparent: the geometry of a space X is represented
by something like a measure on the set of subsets in X; yet, I do not know how to
formulate clear-cut mathematical questions in either case (compare , ).
Who knows where manifolds are going?
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IHES, Bures-sur-Yvette, France
Clay Mathematics Proceedings
Volume 19, 2014
Geometric Analysis on 4-Manifolds
In this expository paper, we will discuss some geometric analytic
approaches to studying the topology and geometry of 4-manifolds. We will
start with a brief summary on 2-manifolds and then recall some aspects of
Perelman’s resolution of the Geometrization conjecture for 3-manifolds by us-
ing Hamilton’s Ricci ﬂow. Then we discuss geometric approaches and progress
on studying 4-manifolds. For simplicity, we assume that all manifolds in this
paper are closed and oriented.
1. Geometrization of 2-manifolds
Let M be a 2-dimensional manifold. Any Riemannian metric g on M gives
rise to a conformal structure and makes M into a Riemann surface. It then follows
from complex analysis that the universal covering of M is conformal to either S
or the hyperbolic disc D. In particular, the topology of M is determined by
its fundamental group. Moreover, since each of the standard spaces above has a
canonical metric with constant curvature, we can conclude that there is a metric
g with constant curvature and conformal to g. In fact, such a ˜
g is unique if the
volume is normalized.
Another approach to studying 2-manifolds is to construct metrics with constant
curvature by solving partial diﬀerential equations. This is more analytic and opens
the possibility of generalization to higher dimensions. Given a Riemannian metric
g on M , consider a new metric ˜
g = e
g for some smooth function. A simple
computation shows that ˜
g has constant curvature μ if and only if
−Δϕ + K(g) = μe
where K(g) denotes the curvature of g and Δ is the Laplacian operator of g. This
equation has been studied a lot: see Chapter 6 of [Au] for a detailed discussion.
Here we give a summary for the readers’ convenience. If μ = 0, it is a linear
equation and has a solution by the standard theory. If μ < 0, by the Maximum
Principle, there is a uniform L
-bound on ϕ. Standard elliptic theory can then be
used to derive a prior bounds on all the derivatives on any solutions of the above
equation; consequently, one can establish existence. When μ > 0, the problem
is more tricky and is often referred as the Nirenberg problem. Many prominent
mathematicians, including Nirenberg, Kazdan–Warner, Aubin et al. studied this
problem. It has been shown that (1.1) always has a solution (see Section 4 of
Supported partially by NSF grant DMS-0804095.
c 2014 Gang Tian
Chapter 6 in [Au]). Therefore, given any g, there is a metric ˜
g conformal to g and
with constant curvature. A classical uniformization theorem in diﬀerential geometry
(cf. Chapter 8, [DoC]) then implies that modulo scaling, the universal covering
of M with the induced metric from ˜
g is isometric to S
or the hyperbolic
disc D with the standard metric. The advantage of this approach is that one gets
a full understanding of geometry and topology of 2-manifolds by solving a partial
A more recent method of ﬁnding metrics with constant curvature is to use the
Ricci ﬂow introduced by R. Hamilton [Ha82]:
−2Ric(g), g(0) = g
In [Ha88], [Ch90], it was proven that given any initial g
, (1.2) has a global solution
g(t) after normalization and g(t) converges to a metric g
on M . One can show
is of constant curvature. The proof is trivial if the Euler number of M is
non-positive and is contained in [CLT06] if M has positive Euler number. Thus
the Ricci ﬂow gives rise to another approach to geometrizing 2-manifolds.
2. Geometrization of 3-manifolds
Can one extend what we said about surfaces to higher dimensions? First we
need to introduce the notion of Einstein metrics.
2.1. g is Einstein if Ric(g) = λg, where λ =
−(n − 1), 0, n − 1.
Note that Ric(g) = (R
) denotes the Ricci curvature of g. It measures the
deviation of volume form from the Euclidean one. In dimension 2, an Einstein
metric is simply a metric with constant Gauss curvature.
Now assume that M is a compact 3-manifold. In this case, an Einstein met-
ric has constant sectional curvature, and the classical uniformization theorem in
diﬀerential geometry (cf. Chapter 8, [DoC]) then states that if M admits an Ein-
stein metric, then its universal covering is of the form S
(M ) and
denotes the hyperbolic space of dimension 3. Thus, if we can
always construct an Einstein metric, then we have a similar picture for 3-manifolds
as we have for surfaces. However, not every 3-manifold admits an Einstein metric.
One can easily construct such examples, such as Σ
for any surface Σ of genus
greater than 1. This is because its fundamental group is the product of a surface
Z which is neither an abelian group nor the fundamental group of any
hyperbolic compact 3-manifold (cf. [Th97]).
It was known [Kn29] that any closed 3-manifold can be decomposed along
embedded 2-spheres into irreducible 3-manifolds; moreover, such a decomposition
is essentially unique. Thurston’s Geometrization Conjecture claims (cf. [Th97],
[CHK00]) that any irreducible 3-manifold can be decomposed along incompressible
tori into ﬁnitely many complete Einstein 3-manifolds plus some Graph manifolds.
The famous Poincare conjecture is a special case of this Geometrization Conjecture.
This conjecture has been solved by Perelman (cf. [Per02], [Per03]) using the
Ricci ﬂow introduced by R. Hamilton in early 80’s:
g(0) = a given metric.
GEOMETRIC ANALYSIS ON 4-MANIFOLDS
R. Hamilton and later DeTurck proved that for any initial metric, there is a unique
solution g(t) on M
×[0, T ) for some T > 0. R. Hamilton also established an analytic
theory for Ricci ﬂow.
If the Ricci ﬂow has a solution g(t), then we can choose a scaling λ(t) > 0 and a
reparametrization t = t(s) with λ(0)
− 1 and t(0) = 0 such that ˜g(x) = λ(s)g(t(s))
has ﬁxed volume and satisﬁes the normalized Ricci ﬂow:
If the normalized Ricci ﬂow (2.2) has a global solution ˜
g(s) for all s
≥ 0 and ˜g(s)
converges to a smooth metric g
as s goes to
∞, then its limiting metric g
Einstein metric, so the universal covering of M is standard. The Geometrization
The ﬁrst successful case was done by R. Hamilton in 1982: If M has a metric of
positive Ricci curvature, then the normalized Ricci ﬂow has a global solution which
converges smoothly to a metric of constant positive curvature, consequently, M is
a quotient of S
by a ﬁnite group.
However, in general, the Ricci ﬂow develops a singularity at ﬁnite time. The
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