Poincaré Conjecture and Geometrization Gang Tian (Peking University)

Clay Millenium Prizes At the first ICM in Paris in 1900, D. Hilbert posed a list of 23 problems from a broad range of mathematical fields. Much of the progress in 20th century mathematics has revolved around these problems. In 2000, inspired by this tradition by Hilbert, the Clay Math Institute identified 7 important and central problems of mathematics and offered $1,000,000 for the solution of each of these problems.

H. Poincaré (1854－1912): One of the greatest French mathematicians. In 1904, Poincaré proposed the famous Poincaré Conjecture which puzzled mathematicians around the world for a hundred years.

Topology is one of core mathematical fields，but it differs from the Euclidean geometry. Topology studies invariant properties under continuous deformation, particularly, quantities, such as length, width, size, area, volume, are not concerned in topology. Poincare conjecture is one of the most famous problems in topology. It gives a topological characterization of the simplest 3-dimensional space, i.e., the 3-sphere. More precisely, Poincare Conjecture states: Any simply connected closed 3-manifold must be homeophic to a round 3-sphere.

For more than one hundred years it has been a driving force in the development of topology to study the Poincaré conjecture, including topological classification of high dimensional spaces in 60s, the study of differentiable structures on 4-dimensional manifolds. Many fundamental questions remain to be open. Low dimensional topology is still a very active research area.

big progresses in topology. It led to a number of big progresses in topology. In 1960, S. Smale generalized it to arbitrary diem and solved the generalized version for dimension 5 or up. In 1982，M. Freedman solved the generalized Conjecture in dimension 4. In 1980, W. Thurston proposed the Geometrization Conjecture on 3-manifolds. Poincaré Conjecture is a corollary of the Geometrization Conjecture. He verified his conjecture for a large class of 3 - manifolds. However, this class of 3- manifolds does not contain the Poincare Conjecture. W. Thurston S. Smale M. Freedman

How to `visualize’ the 3-sphere？ Poincare Conjecture gives a topological characterization of 3-sphere What are special properties of 3-sphere? It is not possible to directly see the 3-dimensional sphere since it does not live in 3-dimensional space. We can understand properties that it has by analogy with what is true for the 2-sphere. So,… how do we think about the 2-sphere?

Stereographic Projection Project 2-sphere with north pole deleted to Euclidean 2-plane. Therefore, 2-sphere can be identified with the Euclidean 2-plane plus an infinity point which corresponds to the north pole. Similarly, we can use mathematical means to get a 3-dimensional stereographic projection from 3-sphere to Euclidean 3-space. Therefore, 3- sphere can be identified with 3-dimensional Euclidean space plus an infinity point. This can be proved in a mathematical rigor.

2-sphere as a union of two disks In view of topology, 2-sphere can be obtained by gluing two disks along their boundary. Therefore, we can imagine that 3-sphere can be obtained by gluing two 3-balls along their boundary which is a 2-sphere.

Two descriptions of 3-sphere 3-dimensional Euclidean space plus a point at infinity Gluing entire boundary together

What is simply-connected? Surfaces and Circles

2-sphere is simply-connected since every loop on it shrinks to a point, that is, 2-sphere has no hole.

3-sphere is simply-connected Like 2-sphere, 3-sphere is simply-connected, i.e., 3-sphere has no hole, every loop on it can shrink to a point. We can prove this simply-connectedness by using the first characterization of 3-sphere. Given any loop on 3-sphere, remove one point outside given loop, the rest can be identified with Euclidean 3-space which has no holes. Hence, given loop can shrink to a point, that is, 3-sphere is simply-connected.

How to prove the Poincare Conjecture? We just proved that 3-sphere is simply-connected , i.e., no hole. Intuitively, the Poincaré Conjecture says that the converse is also true: If the 3-D space has no `holes’ that you can wrap a loop around, then it must be the 3-sphere. For one hundred years, how did topologists try to prove Poincare Conjecture? What is the traditional approach?

Let us start with surfaces The surface is 2-dimensional because (locally at least) one can describe where one is on the surface by giving two numbers. Each of these surfaces sits in 3-D space and is the boundary of a 3-dimensional object, called a solid handlebody. The genus of the surface and of the solid handlebody is the number of holes.

In mathematics, it is called a genus 0 handle-body decomposition. The second characterization of 3-sphere shows that it can be obtained by gluing two balls along their boundary. In mathematics, it is called a genus 0 handle-body decomposition.

But the decomposition is not unique！ In fact, a theorem in mathematics states that every 3-space without boundary (and of finite extent) can be represented by taking two copies of a solid handlebodies of some genus (number of holes) and gluing the entire boundaries together. This is called a handlebody decomposition of the 3-space; its genus is the genus of the handlebodies used. A given 3-space may have handlebody decompositions of different genera. By gluing red latitude circles on one torus green longitude circles on another torus, one can get a genus one handlebody decomposition of 3-sphere.

3-sphere may have handlebody decomposition of higher genus. More precisely, we can start with the genus 0 handlebody decomposition of 3- sphere and follow the steps shown on the right to construct a genus 1 handlebody decomposition of 3-sphere.

3-sphere may have handlebody decomposition of higher genus, say 2. Glue them along entire boundary

Searching for methods of solving the Poincare Conjecture The structure of surfaces of high genus is more complicated than that of 2-sphere, and there are many different ways of gluing them together, so we may get different 3-dimensional spaces. Searching for methods of solving the Poincare Conjecture However, there is an unique way of gluing 2-sphere to a copy of itself, it implies that any 3-space which admits a genus 0 handlebody decomposition must be homeomorphic to the 3-sphere. Therefore, if we can find a method which can decompose a simply-connected 3-space into two 3-dimensional balls, then it must be the 3-sphere. For one hundred years, topologists tried without success to find such a method.

Mathematicians who had tried on Poincare Conjecture The earliest famous mathematician who studied Poincare Conjecture may have been J. Whitehead. In1930，he claimed to have a proof, soon after, he found a mistake in his proof and withdrew it. However, in this process, he found an interesting example of a simply-connected noncompact 3-space which is not homeomorphic to 3-dimensional Euclidean space. In 50s and 60s of last century, many famous mathematicians, e.g., Bing、Haken、Papakyriakopoulos, had tried to solve the conjecture without success.

Christos Papakyriakopoulos Papakyriakopoulos was a Greek mathematician. He came to Princeton in 1948 and then stayed to work there. He proved Dehn Lemma，Closed Loop Theorem and Sphere Theorem. These are fundamental works in 3- dimensional topology. He was awarded the first Veblen Prize in Geometry in 1964. Since early 60s, Papa started to study the Poincare Conjecture until he died of cancer in 1976. He spent almost all his time in office to study the conjecture while listening to music by Wagner. Christos Papakyriakopoulos

Riemannian Geometry Solution of the Poincare Conjecture relies on methods from differential geometry and geometric analysis. It uses Riemannian metrics and curvature.

Riemannian Metrics On any space, one can impose a structure for measuring angles and lengths. Such a structure is called a Riemannian metric. But, there are an infinitely huge number of these Riemannian metrics and no obvious way to construct one with prescribed properties.

Curvature measures how space is curved. It is intrinsic. What is curvature? Curvature measures how space is curved. It is intrinsic.

In higher dimensions In higher dimensions, curvature is much more complicated: every 2-d direction has curvature. Riemann showed that, locally at least, this is the only quantity intrinsically associated with the metric. For example, a small piece of an n-dimensional space can be flattened to a piece of Euclidean space if and only if its curvature is 0. There is a related, simplified curvature, called the Ricci curvature which measures deviation of volume forms on curved spaces from the Euclidean one.

Thurston’s Geometrization Conjecture Thurston’s conjecture says that 3-dimensional spaces can be cut up in a natural way into pieces that admit especially nice metrics, i.e., metrics with constant Ricci curvature. This suggests a different approach to the Poincaré Conjecture and indeed to finding and listing all 3-dimensional spaces– construct the nice Riemannian metric by analytic and differential geometric methods.

In 1982, R. Hamilton introduced the Ricci flow: Hamilton’s Ricci flow is to let the metric evolve by requiring that the time derivative of the metric is proportional to the Ricci curvature. If this flow had a global solution which converges, up to scaling, a metric, then we would prove the Geometrization Conjecture. This is a non-linear equation. It develops singularity.

Richard Hamilton （PhD, 1966, Princeton) Contributions of R. Hamilton R. Hamilton established an analytic foundation for Ricci flow and proposed a program towards solving the Geometrization Conjecture of Thurston for 3-manifolds. He also solved the Poincare Conjecture under certain curvature conditions, but he was unable to overcome some crucial difficulties. Richard Hamilton （PhD, 1966, Princeton)

On Tue, 12 Nov 2002, Grigori Perelman wrote: Dear xxx, may I bring to your attention my paper in arXiv math.DG 0211159 …… Best regards, Grisha Perelman On Nov. 12 of 2002 Perelman posted a preprint online and emailed a number of mathematicians around the world to tell his new paper. In subsequent half a year, he posted another two preprints. In these 3 preprints, he outlined proofs for the Poincare Conjecture as well as Thurston’s Geometrization Conjecture.

The non-linearity of Ricci flow allows singularities to develop and understanding these singularities was the hang-up in completing Hamilton’s program. A crucial advance of Perelman is to prove that finite-time singularities can form only along 2-sphere, thus the change in topology can be completely understood.

After understanding topological nature of singularities, Perelman was able to construct a generalized solution for Ricci flow, often referred as Ricci flow with surgery, for all time and any initial data. Using the min-max principle, Perelman outlined a proof for that the Ricci flow with surgery becomes extinct in finite time. Then the Poincare Conjecture follows.

Perelman did not provide all the details others hoped for his solution，this made it very hard for others to understand his proof. Through efforts of several mathematicians for two years, we eventually filled in necessary details for the proof of Poincare Conjecture. Though there were small gaps, they could be easily fixed.

The solution of Perelman used many important advances in differential geometry in last 50 years: Classification of non-negative curved spaces; Compactness in Riemannian Geometry; Harnack-type estimates for heat equations; Collapsing theory for spaces with curvature bounded from below; Theory of minimal surfaces…….

The proof of Geometrization Conjecture needs more efforts. In 2012, R. Bamler found an unsolved technical issue in the proof of Perelman and neglected by those who provided a detailed proof. Luckily, by exploiting Perelman’s method, Bamler solved this problem. Furthermore, he proved a refined version of Geometrization Conjecture. R. Bamler

An unsolved problem Though M. Freedman solved the 4-dimensional generalized Poincare Conjecture, we do not know if there is an exotic 4-sphere, i.e., a smooth space which is homeomorphic but not diffeomorphic to the standard 4-sphere. This is called Smooth 4-d Poincare Conjecture, which is still open. In 1957，J. Milnor found 28 exotic 7-spheres which not diffeomorphic to standard 7-sphere.

Is there a differential geometric method to solve the smooth 4-d Poincare Conjecture, such as a geometric flow which plays a role as Ricci flow does in dimension 3? It is known that no exotic 4-sphere admits self-dual metrics. Is there a geometric flow for solving self-dual equation in dimension 4? Symplectic 4-manifolds play a very important role in studying 4-manifolds. J. Streets and I introduced a new curvature flow which preserves symplectic structure.

Analytic Minimal Model Program was initiated by myself together with my collaborators. It aims at classifying algebraic manifolds. It uses the Ricci flow. The main obstacle is to understand finite-time singularities which correspond to flips in algebraic geometry.

Thanks!