Highlights in Differential Geometry: 1950 to 2011 Lecture 1: Introduction Richard Schoen Stanford University

S. S. Chern Lecture Series, Beijing, China September 22, 2011

Plan of Lecture The lecture will have four parts: Part 1: History and a general description of the subject. Part 2: The Weyl embedding problem. Part 3: Geometric variational problems; Plateau and Yamabe problems. Part 4: Geometric flows; Ricci and mean curvature flow. This lecture is not intended to be a list of the most important developments in the subject since 1950. The selection is based on my interests. I have not mentioned complex geometry, so have omitted Yau’s solution of the Calabi Conjecture and the subsequent development in String Theory. I have also omitted the Atiyah-Singer Index Theorem. Both of these belong on any list of the most important developments.

Part 1: Brief History of Differential Geometry Differential Geometry begins with curves in the plane. Greek geometry, as exposited in Euclid’s Elements, focused on lines and circles in the plane and space.

Curvature of Curves

Curves whose curvature varies from point to point entered into geometry and physics beginning in the 14th century with important contributions by N. Oresme, J. Kepler, I. Newton, and L. Euler.

The curvature at a point is the reciprocal of the radius of the circle that best fits the curve at the point.

Curves Euler developed much of the modern theory of plane curves.

Curved Surfaces I A surface is an infinitely thin two dimensional membrane in space.

M¨obius Band

Torus

Higher Genus Surfaces

Curved Surfaces II The key defining property of a surface is that we can introduce (two) local coordinates near any point of a surface.

Geometry of Surfaces I The Differential Geometry of surfaces was developed by C. F. Gauss around 1825.

Geometry of Surfaces II Gauss defined the curvature of a surface in terms of curvatures of curves on the surface.

Intrinsic Geometry Certain aspects of the geometry of a surface are determined by measurements made along the surface without reference to the larger three dimensional space. Such quantities are called intrinsic, and these include lengths of curves on the surface, angles between curves, and areas of regions on the surface. Gauss’ most remarkable discovery, which he called his Theorema Egregium, is the result that a certain expression in the curvatures of a surface is intrinsic. This expression is K = k1 k2 , and it is called the Gauss curvature of the surface. The proof involved a very difficult calculation. This motivated the study of intrinsic geometries or metrics on a surface which do not necessarily come from embeddings into three space. These are measurements of distances and angles for which the Pythagorean theorem holds approximately for small right triangles.

Higher Dimensions The intrinsic geometry of Gauss was generalized to higher dimensions by B. Riemann in 1854.

Sectional Curvature Just as the curvatures of a surface can be understood in terms of curvatures of certain curves on the surface, the curvatures of higher dimensional spaces (called Riemannian manifolds) can be understood in terms of the Gauss curvatures of certain surfaces passing through a point. These Gauss curvatures are called sectional curvatures and there is one for each two dimensional plane at a point.

Part 2: The Weyl Embedding Problem I

A two dimensional geometry may not be embedded in a natural way in space. For example, the hyperbolic plane is a geometry of Gauss curvature equal to −1, and while pieces of it can be embedded in space, a theorem of Hilbert asserts that it cannot be globally realized in space. In 1916 H. Weyl conjectured and outlined a proof that a closed surface of positive Gauss curvature can be isometrically embedded in space as the boundary of a convex body. He was not able to complete the proof because the required analysis was not available at that time. H. Lewy completed the proof in a special case (real analytic geometry) in 1938.

Convex Surfaces. We have seen a picture of a closed surface of positive curvature.

Weyl Embedding Problem II H. Weyl was a great mathematician and physicist who lived from 1885 to 1955.

General Idea of Proof I Weyl showed that any geometry S1 of positive curvature on the sphere can be joined to the standard unit sphere S0 by a family of geometries St for 0 ≤ t ≤ 1 all having positive curvature. We know that S0 can be embedded as the unit sphere, and Weyl showed that St can be embedded for very small t.

General Idea of Proof II The hope is to show that this process can be continued until t = 1, and we get an embedding of the geometry of interest. The main difficulty in showing this is to show that the embeddings of St remain smooth and do not crease or develop some type of singularity. This requires estimates for nonlinear partial differential equations which were not available in 1916. The necessary estimates were developed by C. B. Morrey in 1938 and by L. Nirenberg in 1953. These led to a complete solution of the Weyl problem by Nirenberg in 1953. An alternative approach was developed independently by A. V. Pogorelov. This is an early example of a problem in global differential geometry which was solved by analytic methods.

Part 3: Geometric Variational Problems It is common in geometric problems that we hope to choose a nice geometric object from infinitely many possibilities. Many natural and important geometric objects are characterized as minimizers or extremals of some ‘size’ or energy measure. For example there are infinitely many possible geometries on the sphere.

How can the round geometry be characterized among the others?

The Plateau Problem Perhaps the most visually appealing geometric variational problem is the Plateau problem, or the problem of finding a surface of smallest area spanning a given curve. Such surfaces occur in nature as soap films and they achieve beautiful shapes.

Minimal Surfaces I The surfaces which arise as surfaces of extremal area are called minimal surfaces. These have been studied for hundreds of years. Here are a few examples.

This surface is called the catenoid.

Minimal Surfaces II This is the helicoid.

Minimal Surfaces III This is a minimal surface discovered by H. A. Schwarz in the 19th century.

Mathematical work on the Plateau Problem The Plateau problem was formulated by Lagrange in 1760, but the first rigorous solution was given in 1930 by J. Douglas and T. Rado. It dealt only with two dimensional surfaces with a fixed topology (such as a disk). Douglas received the first Fields Medal in 1936 for his work. The methods did not extend to the higher dimensional Plateau problem. There was spectacular development on the higher dimensional Plateau problem beginning in the late 1950s. This was achieved through work of many authors including deGiorgi, Federer, Fleming, Reifenberg, Almgren, Bombieri, Giusti, and Simons. Powerful methods were developed to obtain partial regularity/smoothness of solutions in situations where singularities do occur. These methods have had a very big impact in modern differential geometry.

Minimal Surfaces in Curved Spaces One dimensional minimal surfaces are geodesics, and the study of geodesics on a surface or a higher dimensional space can be used to understand the geometry of the space.

Minimal Surfaces and Black Holes In general relativity we study four dimensional curved spacetimes. If we imagine a time function and look at a fixed time slice then we get a three dimensional curved space. The existence of a closed minimal surface in that space is an indication of a black hole forming in the spacetime. The reason for this is that being minimal means that the mean curvature is zero. The mean curvature of the surface is a measure of the focusing of light rays emanating from points of the surface. Because of the Einstein equations it is true that light rays which are initially focusing will continue to focus, and it follows that points of the surface cannot escape to infinity and the spacetime must be singular (Penrose Theorem). Such surfaces are called trapped, and the existence of a minimal surface which is locally area minimizing indicates the existence of trapped surfaces nearby. The minimal surface is called an apparent horizon.

The Schwarzschild Black Hole

The Positive Mass Theorem One of the major applications of minimal surface theory was to prove the Positive Mass Theorem in general relativity. This was a problem going back to the beginnings of relativity, and was solved by the speaker and S. T. Yau in 1978. The intuitive idea of this result is that a general solution of the Einstein equations which represents a finite gravitating system will look from a distance like a Schwarzschild solution. The total (ADM) mass of the system is then defined to be the mass of that Schwarzschild solution. The conjecture was that this mass should always be nonnegative. This implies that there will be a net attraction between two distant gravitating systems, and thus the net gravitational force in general relativity is attractive. If this were not the case, it would imply an instability in the theory of general relativity which would not be physically acceptable.

Rough Idea of the Proof The starting point of the proof is the statement that apparent horizons (minimal surfaces) must be spherical. This is a consequence of the Einstein equations and goes back to Gibbons and Hawking, and was generalized by the speaker and Yau. We showed that if there existed a spacetime with negative mass then it would be possible to construct a minimal surface inside which is asymptotic to a plane, a contradiction. The proof and its higher dimensional generalization uses the full strength of the existence and regularity theory for the Plateau problem.

Constant Curvature Metrics We began our discussion of variational problems by asking how we could distinguish a constant curvature geometry on the sphere from the other infinitely many geometries. This is classical on the two dimensional sphere.

In 1960 Yamabe introduced a variational approach for doing something similar for three dimensional geometries. The energy functional he considered is a very natural one called the Hilbert-Einstein energy. It is the energy which leads to the Einstein equations.

The Yamabe Problem The minimizers that Yamabe proposed to construct would have constant scalar curvature, and would be gotten by doing conformal (angle preserving) deformations of a given geometry. Yamabe thought he had solved the problem, and he passed away before his 1960 paper appeared. It was pointed out by N. Trudinger in 1968 that he had neglected a certain type of divergence which occurs in similar problems. The Yamabe Conjecture then became a much studied problem in differential geometry. It was not until 1984 that the speaker was able to construct the minimizers for this problem. Earlier work was done by T. Aubin who resolved some cases but not the original three dimensional case studied by Yamabe. It turns out that the problem has many intricacies, and in different dimensions the results differ. Nevertheless it has been shown that at least one minimizer exists on closed geometries in all dimensions.

High Energy Critical Points for the Yamabe Problem There are many cases for the Yamabe problem in which, besides minimizers, there are very high energy critical points. These are constant scalar curvature geometries conformal to the given one with high energy. Here is a picture of such a solution on the three dimensional space S 1 × S 2 . You should think of the ends as being identified.

Rough Idea of the Proof The goal in the Yamabe problem is to prevent the blow up of solutions. Since such blow up occurs in closely related problems, there is no hope of doing this from the local structure of the equation. The blow up can only be prevented by finding some regularizing effect from the geometry. For minimizing solutions, T. Aubin found such an effect for generic metrics on manifolds of dimension six or greater. The speaker found a new connection of this blow up with general relativity and was able to use the positive mass theorem to prevent blow up of minimizing solution in all remaining cases. The problem of preventing blow up for solutions which are not minimizing turns to be much more subtle. It was done in the cases for which the speaker solved the Yamabe problem in the late 1980s, but the general case was not handled until recently. Through work of S. Brendle, M. Khuri, F. Marques, and the speaker we now know that solutions do blow up for some metrics on manifolds of dimension 25 or higher, while for all (non-spherical) metrics of dimension less than 25 no blow up can occur.

Part 4: Geometric Flows, Some History If one seeks an equilibrium configuration such as a minimal surface, it is often possible to formulate an evolution equation whose stationary points are the equilibria. The model is the equation of heat diffusion in a conducting plate where the heat equation takes any initial heat distribution and evolves so that it converges rapidly to an equilibrium. The basic principle is that regions of higher temperature tend to cool at a specific rate, and this leads to a differential equation whose solution is the temperature at a given point on the plate at a given time. It is also natural to study heat diffusion on a Riemannian manifold. In this case the geometry of the manifold influences the behavior of the solution. The quantitative study of this relationship was pioneered by P. Li and S. T. Yau who proved basic results on the behavior of solutions including the Harnack inequality which gives a sharp relationship between the temperatures at two different points in space and time which depends on the curvature of the space and the separation of the points in spacetime.

Some History of the Heat Equation in Geometry The heat equation was used in differential geometry by A. Milgram and P. Rosenbloom in 1951 to prove the Hodge theorem, which involves the solution of a linear equation. In 1964 there was an important theorem of J. Eells and J. Sampson on the harmonic map heat flow. In both cases it was possible to deform an initial configuration to equilibrium without singularities or discontinuities. In general, solutions of nonlinear heat equations do not evolve smoothly, but rather develop singular behavior meaning that the solution or some derivative blows up after a finite amount of time. Understanding the nature of this blow up is a basic problem for these equations. For geometric equations this understanding is often essential for geometric applications.

Mean Curvature Flow The mean curvature flow is the evolution problem which prescribes that a surface moves at each point in the normal direction with a speed equal to the mean curvature. This is a nonlinear heat equation whose fixed points are minimal surfaces. It is the flow which reduces the area of the surface as fast as possible. In most cases the mean curvature flow develops singularities after some finite time. The first systematic mathematical treatment of mean curvature flow was done in 1978 by K. Brakke. The equations appeared earlier in the materials science literature. Because of the influence of R. Hamilton and S. T. Yau, flows became a much more active field of study in geometry after 1980. Since then, there has been very substantial progress made on mean curvature flow, but many questions remain open and it is a very active field of investigation.

Flow for curves In 1987 M. Grayson showed that any embedded plane curve flows to a circle under mean curvature flow.

Flow for Surfaces For surfaces the situation is very different in general. Many questions remain concerning mean curvature flow for surfaces.

Ricci Flow The study of geometric flows became a central area of differential geometry after 1980 with R. Hamilton’s invention of the Ricci flow. It is a flow of intrinsic geometries or Riemannian metrics. It specifies that the rate of change of the metric be given by a specific part of the Riemann curvature tensor called the Ricci curvature. It turns out to be a heat-type equation, so even though it is nonlinear, it can be solved for a short time. Hamilton used the Ricci flow to prove a remarkable new theorem in Riemannian geometry. He showed that any closed three dimensional space of positive curvature is either a three dimensional sphere, or one of a known list of spaces closely related to the three sphere. This theorem established the Ricci flow as an important method to apply to a variety of problems which involve finding special geometries.

Three Dimensions I The most spectacular application of the Ricci flow is to the topology of three dimensional manifolds. For example, the Poincar´e conjecture says that the only closed, simply connected, three dimensional manifold is the three dimensional sphere.

Three Dimensions II

From 1980 onward, Hamilton’s main goal for the Ricci flow was to use it to understand the topology of three manifolds. He realized that a sufficiently good understanding of a general three dimensional Ricci flow including the singular behavior would be sufficient. With the addition of the spectacular work of G. Perelman in 2002 this goal has been achieved. The detailed proofs remain very complicated though.

The Differentiable Sphere Theorem Another application of the Ricci flow is to high dimensional geometry. Hamilton showed that a closed three manifold of positive curvature is a sphere (or related space), but this result is not true in dimensions greater than three. Attempts to characterize the sphere by curvature go back to H. Hopf before 1950. The sharp condition which was posed is called 1/4-pinching, and it means that the largest sectional curvature is less than 4 times the smallest. The conjecture that a 1/4-pinched manifold is a sphere (or related space) is called the differentiable sphere theorem. Since 1950 partial results were obtained by many mathematicians including H. Rauch, W. Klingenberg, and M. Berger. The full conjecture was finally solved using the Ricci flow by S. Brendle and the speaker in 2007.

Rough Idea of the Proof The basic idea of the proof of the differentiable sphere theorem is to start the Ricci flow with a 1/4-pinched metric and to show that the (normalized) flow evolves without singularity to a constant curvature metric. The problem is that the assumption of 1/4-pinching is not preserved under the flow. The way we get around this problem is to find a larger set of metrics which is invariant under the flow, but is small enough that singularities can be excluded and that the evolution makes the metrics more round, so that the equilibrium limit is a constant curvature metric. This shows that the manifold has a constant curvature metric and is therefore either the sphere or a related space. Important earlier work on invariant sets of metrics was done by Hamilton and improved substantially by C. B¨ ohm and B. Wilking.

S. S. Chern Lecture Series, Beijing, China September 22, 2011

Plan of Lecture The lecture will have four parts: Part 1: History and a general description of the subject. Part 2: The Weyl embedding problem. Part 3: Geometric variational problems; Plateau and Yamabe problems. Part 4: Geometric flows; Ricci and mean curvature flow. This lecture is not intended to be a list of the most important developments in the subject since 1950. The selection is based on my interests. I have not mentioned complex geometry, so have omitted Yau’s solution of the Calabi Conjecture and the subsequent development in String Theory. I have also omitted the Atiyah-Singer Index Theorem. Both of these belong on any list of the most important developments.

Part 1: Brief History of Differential Geometry Differential Geometry begins with curves in the plane. Greek geometry, as exposited in Euclid’s Elements, focused on lines and circles in the plane and space.

Curvature of Curves

Curves whose curvature varies from point to point entered into geometry and physics beginning in the 14th century with important contributions by N. Oresme, J. Kepler, I. Newton, and L. Euler.

The curvature at a point is the reciprocal of the radius of the circle that best fits the curve at the point.

Curves Euler developed much of the modern theory of plane curves.

Curved Surfaces I A surface is an infinitely thin two dimensional membrane in space.

M¨obius Band

Torus

Higher Genus Surfaces

Curved Surfaces II The key defining property of a surface is that we can introduce (two) local coordinates near any point of a surface.

Geometry of Surfaces I The Differential Geometry of surfaces was developed by C. F. Gauss around 1825.

Geometry of Surfaces II Gauss defined the curvature of a surface in terms of curvatures of curves on the surface.

Intrinsic Geometry Certain aspects of the geometry of a surface are determined by measurements made along the surface without reference to the larger three dimensional space. Such quantities are called intrinsic, and these include lengths of curves on the surface, angles between curves, and areas of regions on the surface. Gauss’ most remarkable discovery, which he called his Theorema Egregium, is the result that a certain expression in the curvatures of a surface is intrinsic. This expression is K = k1 k2 , and it is called the Gauss curvature of the surface. The proof involved a very difficult calculation. This motivated the study of intrinsic geometries or metrics on a surface which do not necessarily come from embeddings into three space. These are measurements of distances and angles for which the Pythagorean theorem holds approximately for small right triangles.

Higher Dimensions The intrinsic geometry of Gauss was generalized to higher dimensions by B. Riemann in 1854.

Sectional Curvature Just as the curvatures of a surface can be understood in terms of curvatures of certain curves on the surface, the curvatures of higher dimensional spaces (called Riemannian manifolds) can be understood in terms of the Gauss curvatures of certain surfaces passing through a point. These Gauss curvatures are called sectional curvatures and there is one for each two dimensional plane at a point.

Part 2: The Weyl Embedding Problem I

A two dimensional geometry may not be embedded in a natural way in space. For example, the hyperbolic plane is a geometry of Gauss curvature equal to −1, and while pieces of it can be embedded in space, a theorem of Hilbert asserts that it cannot be globally realized in space. In 1916 H. Weyl conjectured and outlined a proof that a closed surface of positive Gauss curvature can be isometrically embedded in space as the boundary of a convex body. He was not able to complete the proof because the required analysis was not available at that time. H. Lewy completed the proof in a special case (real analytic geometry) in 1938.

Convex Surfaces. We have seen a picture of a closed surface of positive curvature.

Weyl Embedding Problem II H. Weyl was a great mathematician and physicist who lived from 1885 to 1955.

General Idea of Proof I Weyl showed that any geometry S1 of positive curvature on the sphere can be joined to the standard unit sphere S0 by a family of geometries St for 0 ≤ t ≤ 1 all having positive curvature. We know that S0 can be embedded as the unit sphere, and Weyl showed that St can be embedded for very small t.

General Idea of Proof II The hope is to show that this process can be continued until t = 1, and we get an embedding of the geometry of interest. The main difficulty in showing this is to show that the embeddings of St remain smooth and do not crease or develop some type of singularity. This requires estimates for nonlinear partial differential equations which were not available in 1916. The necessary estimates were developed by C. B. Morrey in 1938 and by L. Nirenberg in 1953. These led to a complete solution of the Weyl problem by Nirenberg in 1953. An alternative approach was developed independently by A. V. Pogorelov. This is an early example of a problem in global differential geometry which was solved by analytic methods.

Part 3: Geometric Variational Problems It is common in geometric problems that we hope to choose a nice geometric object from infinitely many possibilities. Many natural and important geometric objects are characterized as minimizers or extremals of some ‘size’ or energy measure. For example there are infinitely many possible geometries on the sphere.

How can the round geometry be characterized among the others?

The Plateau Problem Perhaps the most visually appealing geometric variational problem is the Plateau problem, or the problem of finding a surface of smallest area spanning a given curve. Such surfaces occur in nature as soap films and they achieve beautiful shapes.

Minimal Surfaces I The surfaces which arise as surfaces of extremal area are called minimal surfaces. These have been studied for hundreds of years. Here are a few examples.

This surface is called the catenoid.

Minimal Surfaces II This is the helicoid.

Minimal Surfaces III This is a minimal surface discovered by H. A. Schwarz in the 19th century.

Mathematical work on the Plateau Problem The Plateau problem was formulated by Lagrange in 1760, but the first rigorous solution was given in 1930 by J. Douglas and T. Rado. It dealt only with two dimensional surfaces with a fixed topology (such as a disk). Douglas received the first Fields Medal in 1936 for his work. The methods did not extend to the higher dimensional Plateau problem. There was spectacular development on the higher dimensional Plateau problem beginning in the late 1950s. This was achieved through work of many authors including deGiorgi, Federer, Fleming, Reifenberg, Almgren, Bombieri, Giusti, and Simons. Powerful methods were developed to obtain partial regularity/smoothness of solutions in situations where singularities do occur. These methods have had a very big impact in modern differential geometry.

Minimal Surfaces in Curved Spaces One dimensional minimal surfaces are geodesics, and the study of geodesics on a surface or a higher dimensional space can be used to understand the geometry of the space.

Minimal Surfaces and Black Holes In general relativity we study four dimensional curved spacetimes. If we imagine a time function and look at a fixed time slice then we get a three dimensional curved space. The existence of a closed minimal surface in that space is an indication of a black hole forming in the spacetime. The reason for this is that being minimal means that the mean curvature is zero. The mean curvature of the surface is a measure of the focusing of light rays emanating from points of the surface. Because of the Einstein equations it is true that light rays which are initially focusing will continue to focus, and it follows that points of the surface cannot escape to infinity and the spacetime must be singular (Penrose Theorem). Such surfaces are called trapped, and the existence of a minimal surface which is locally area minimizing indicates the existence of trapped surfaces nearby. The minimal surface is called an apparent horizon.

The Schwarzschild Black Hole

The Positive Mass Theorem One of the major applications of minimal surface theory was to prove the Positive Mass Theorem in general relativity. This was a problem going back to the beginnings of relativity, and was solved by the speaker and S. T. Yau in 1978. The intuitive idea of this result is that a general solution of the Einstein equations which represents a finite gravitating system will look from a distance like a Schwarzschild solution. The total (ADM) mass of the system is then defined to be the mass of that Schwarzschild solution. The conjecture was that this mass should always be nonnegative. This implies that there will be a net attraction between two distant gravitating systems, and thus the net gravitational force in general relativity is attractive. If this were not the case, it would imply an instability in the theory of general relativity which would not be physically acceptable.

Rough Idea of the Proof The starting point of the proof is the statement that apparent horizons (minimal surfaces) must be spherical. This is a consequence of the Einstein equations and goes back to Gibbons and Hawking, and was generalized by the speaker and Yau. We showed that if there existed a spacetime with negative mass then it would be possible to construct a minimal surface inside which is asymptotic to a plane, a contradiction. The proof and its higher dimensional generalization uses the full strength of the existence and regularity theory for the Plateau problem.

Constant Curvature Metrics We began our discussion of variational problems by asking how we could distinguish a constant curvature geometry on the sphere from the other infinitely many geometries. This is classical on the two dimensional sphere.

In 1960 Yamabe introduced a variational approach for doing something similar for three dimensional geometries. The energy functional he considered is a very natural one called the Hilbert-Einstein energy. It is the energy which leads to the Einstein equations.

The Yamabe Problem The minimizers that Yamabe proposed to construct would have constant scalar curvature, and would be gotten by doing conformal (angle preserving) deformations of a given geometry. Yamabe thought he had solved the problem, and he passed away before his 1960 paper appeared. It was pointed out by N. Trudinger in 1968 that he had neglected a certain type of divergence which occurs in similar problems. The Yamabe Conjecture then became a much studied problem in differential geometry. It was not until 1984 that the speaker was able to construct the minimizers for this problem. Earlier work was done by T. Aubin who resolved some cases but not the original three dimensional case studied by Yamabe. It turns out that the problem has many intricacies, and in different dimensions the results differ. Nevertheless it has been shown that at least one minimizer exists on closed geometries in all dimensions.

High Energy Critical Points for the Yamabe Problem There are many cases for the Yamabe problem in which, besides minimizers, there are very high energy critical points. These are constant scalar curvature geometries conformal to the given one with high energy. Here is a picture of such a solution on the three dimensional space S 1 × S 2 . You should think of the ends as being identified.

Rough Idea of the Proof The goal in the Yamabe problem is to prevent the blow up of solutions. Since such blow up occurs in closely related problems, there is no hope of doing this from the local structure of the equation. The blow up can only be prevented by finding some regularizing effect from the geometry. For minimizing solutions, T. Aubin found such an effect for generic metrics on manifolds of dimension six or greater. The speaker found a new connection of this blow up with general relativity and was able to use the positive mass theorem to prevent blow up of minimizing solution in all remaining cases. The problem of preventing blow up for solutions which are not minimizing turns to be much more subtle. It was done in the cases for which the speaker solved the Yamabe problem in the late 1980s, but the general case was not handled until recently. Through work of S. Brendle, M. Khuri, F. Marques, and the speaker we now know that solutions do blow up for some metrics on manifolds of dimension 25 or higher, while for all (non-spherical) metrics of dimension less than 25 no blow up can occur.

Part 4: Geometric Flows, Some History If one seeks an equilibrium configuration such as a minimal surface, it is often possible to formulate an evolution equation whose stationary points are the equilibria. The model is the equation of heat diffusion in a conducting plate where the heat equation takes any initial heat distribution and evolves so that it converges rapidly to an equilibrium. The basic principle is that regions of higher temperature tend to cool at a specific rate, and this leads to a differential equation whose solution is the temperature at a given point on the plate at a given time. It is also natural to study heat diffusion on a Riemannian manifold. In this case the geometry of the manifold influences the behavior of the solution. The quantitative study of this relationship was pioneered by P. Li and S. T. Yau who proved basic results on the behavior of solutions including the Harnack inequality which gives a sharp relationship between the temperatures at two different points in space and time which depends on the curvature of the space and the separation of the points in spacetime.

Some History of the Heat Equation in Geometry The heat equation was used in differential geometry by A. Milgram and P. Rosenbloom in 1951 to prove the Hodge theorem, which involves the solution of a linear equation. In 1964 there was an important theorem of J. Eells and J. Sampson on the harmonic map heat flow. In both cases it was possible to deform an initial configuration to equilibrium without singularities or discontinuities. In general, solutions of nonlinear heat equations do not evolve smoothly, but rather develop singular behavior meaning that the solution or some derivative blows up after a finite amount of time. Understanding the nature of this blow up is a basic problem for these equations. For geometric equations this understanding is often essential for geometric applications.

Mean Curvature Flow The mean curvature flow is the evolution problem which prescribes that a surface moves at each point in the normal direction with a speed equal to the mean curvature. This is a nonlinear heat equation whose fixed points are minimal surfaces. It is the flow which reduces the area of the surface as fast as possible. In most cases the mean curvature flow develops singularities after some finite time. The first systematic mathematical treatment of mean curvature flow was done in 1978 by K. Brakke. The equations appeared earlier in the materials science literature. Because of the influence of R. Hamilton and S. T. Yau, flows became a much more active field of study in geometry after 1980. Since then, there has been very substantial progress made on mean curvature flow, but many questions remain open and it is a very active field of investigation.

Flow for curves In 1987 M. Grayson showed that any embedded plane curve flows to a circle under mean curvature flow.

Flow for Surfaces For surfaces the situation is very different in general. Many questions remain concerning mean curvature flow for surfaces.

Ricci Flow The study of geometric flows became a central area of differential geometry after 1980 with R. Hamilton’s invention of the Ricci flow. It is a flow of intrinsic geometries or Riemannian metrics. It specifies that the rate of change of the metric be given by a specific part of the Riemann curvature tensor called the Ricci curvature. It turns out to be a heat-type equation, so even though it is nonlinear, it can be solved for a short time. Hamilton used the Ricci flow to prove a remarkable new theorem in Riemannian geometry. He showed that any closed three dimensional space of positive curvature is either a three dimensional sphere, or one of a known list of spaces closely related to the three sphere. This theorem established the Ricci flow as an important method to apply to a variety of problems which involve finding special geometries.

Three Dimensions I The most spectacular application of the Ricci flow is to the topology of three dimensional manifolds. For example, the Poincar´e conjecture says that the only closed, simply connected, three dimensional manifold is the three dimensional sphere.

Three Dimensions II

From 1980 onward, Hamilton’s main goal for the Ricci flow was to use it to understand the topology of three manifolds. He realized that a sufficiently good understanding of a general three dimensional Ricci flow including the singular behavior would be sufficient. With the addition of the spectacular work of G. Perelman in 2002 this goal has been achieved. The detailed proofs remain very complicated though.

The Differentiable Sphere Theorem Another application of the Ricci flow is to high dimensional geometry. Hamilton showed that a closed three manifold of positive curvature is a sphere (or related space), but this result is not true in dimensions greater than three. Attempts to characterize the sphere by curvature go back to H. Hopf before 1950. The sharp condition which was posed is called 1/4-pinching, and it means that the largest sectional curvature is less than 4 times the smallest. The conjecture that a 1/4-pinched manifold is a sphere (or related space) is called the differentiable sphere theorem. Since 1950 partial results were obtained by many mathematicians including H. Rauch, W. Klingenberg, and M. Berger. The full conjecture was finally solved using the Ricci flow by S. Brendle and the speaker in 2007.

Rough Idea of the Proof The basic idea of the proof of the differentiable sphere theorem is to start the Ricci flow with a 1/4-pinched metric and to show that the (normalized) flow evolves without singularity to a constant curvature metric. The problem is that the assumption of 1/4-pinching is not preserved under the flow. The way we get around this problem is to find a larger set of metrics which is invariant under the flow, but is small enough that singularities can be excluded and that the evolution makes the metrics more round, so that the equilibrium limit is a constant curvature metric. This shows that the manifold has a constant curvature metric and is therefore either the sphere or a related space. Important earlier work on invariant sets of metrics was done by Hamilton and improved substantially by C. B¨ ohm and B. Wilking.