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The Contributions of Chuu-Lian Terng to Geometry

Karen Uhlenbeck

Communicated by Notices Associate Editor Chikako Mese

1. Introduction

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Chuu-Lian Terng is a geometer who has made important contributions to both submanifold geometry and integrable systems. My collaboration with her over many years has been rewarding and profitable. This article is an attempt to share my appreciation of her work with a wide audience. I am indebted to her for decades of profitable collaboration, a warm personal relationship and help with writing this article.

The article is incomplete, as space constraints do not allow me to discuss all of her research. I chose to emphasize personal details and her work on integrable systems. Many schools of mathematics contributed to the development of integrable systems and I cannot include them all. The outline given in Section 6 is as I have come to see the subject. This is close, but not identical to Terng’s view, and will differ from other views. I hope to give a window opening onto her work, not to provide a definitive treatise. The ideas belong to many people and errors are mine. The reader will surely join me in thanking the referees for helpful comments and important corrections.

2. Background and Education

Chuu-Lian Terng was born in 1949 in Hualian, Taiwan. Her family moved to Taipei when she was three. Her father was in the army under Chiang Kai-shek and had moved from mainland China to Taiwan when the Communists under Mao took over China in 1948. She was the eldest, with three younger brothers, and her family was very poor. She was warned that she had to do well in school, lest she end up working as a maid.

Chuu-Lian was a good student, and graduated the Taipei First Girls’ High School. Her comment: “Mathematics was easy. I liked it.” When it came time to go to college, she was automatically admitted to the National Taiwan University on the basis of her grades. This was not free, unlike the Normal University, but graduates of the Normal University were expected to teach for ten years in middle and high school after graduation. Her sights were set higher. So she tutored middle school and high school students for ten hours a week to earn tuition, books and money to give her family. And she of course lived at home.

Chuu-Lian’s class had 8 women and 24 men. At first it was a big shock for the women to study for the first time with men. In the boys school many students had already studied calculus and read English mathematics books on their own. English was not Chuu-Lian’s strong suit. (More about that later!) Her recollection is that the first year there were two math courses: calculus and “What is Mathematics” from Courant and Robbins. However, the students got together, ran a seminar and studied extra material. There was a lot of homework, and the exams were hard. She recalls that in her senior year she just missed passing an algebra course with a 55 when 60 was passing. Only one student had a higher grade. “The atmosphere was good. The students worked together and presented material, teaching themselves. I remember when I once got the highest grades in my freshmen year, and the men called me ‘brother Terng.’” She graduated at the top of her class.

At that time there were no PhD programs in Taiwan. The top students routinely went to the United States for graduate study and most of them did get PhD’s. They often went to the top graduate schools and did well there. But Chuu-Lian was faced with two problems. The first was her lack of proficiency in English. It was her worst subject and she hated it. All the Taiwanese students had to take the TOEFL exams, and almost all graduate schools in the US required at least a score of 550. Chuu-Lian’s score was 520. She found three schools that only required 500, applied and was accepted into all three. She elected to go to Brandeis.

The second problem was that her family had no money and she had been giving them most of her tutoring earnings. So she delayed for a year, working as a TA and taking a few more courses. This time she kept her earnings for her plane fare and a nest egg until settled at Brandeis. She again sent money home during her graduate student years.

“I loved my courses at Brandeis. I also found out how good my undergraduate training had been.” She still speaks with enthusiasm of Ed Brown teaching topology, Maurice Auslander who gave lot of homework and no lectures and Dick Palais’s course on pseudo-differential operators on the circle. And she remembers what many of us discovered: that we loved complex variables, but that the charm does not necessarily carry over to several complex variables. Her subjects were geometry and topology. No PDE and bad memories of a course in harmonic analysis. At the end of the first year she started working with Dick Palais. Her first readings were his book on the Atiyah-Singer index theorem and Spivak’s Differential Topology volume I. Her thesis was on the classifications of natural vector bundles and natural differential operators.

3. Early Career

Terng’s first postdoctoral position was at the University of California at Berkeley, where she went to study differential geometry with S. S. Chern. When she went to ask him for a problem, his reply was “I don’t give problems. Go to seminars, library, talk to people and find your own problems.” Then Chern became interested in solitons and Bäcklund transformations.

3.1. The sine Gordon equation and Bäcklund transformations

The sine Gordon equation is the equation for ,

We easily see that a trivial solution is given by . We use the notation that subscripts denote partial derivatives.

Theorem 3.1.1.

If is a solution of the sine-Gordon equation and

for any constant , then is also a solution of the sine Gordon equation.

In fact, we see that the two equations for are compatible only when solves the sine Gordon equation. Just as a test, if is the trivial solution, then, we get

Changing variables to and gives equations and , yielding a one-parameter family of solutions. The process can be iterated, giving a chain of solutions known as solitons. After two steps there are relations between solutions known as permutability formulas. We have found a chain of special solutions to the partial differential equation by solving systems of ordinary differential equations. This is the purpose of Bäcklund transformations.

The three classical “integrable systems” with Bäcklund transformations, scattering and inverse scattering theory, tau functions and Virasoro actions are:

(i)

sine Gordon equation (SGE)

(ii)

nonlinear Schrödinger equation (NLS)

(iii)

Korteweg-de Vries equation (KdV)

These three equations have similar properties but very different origins. The sine Gordon equation arose in 1862 in the investigation by Edmund Bour into constant Gaussian curvature surfaces in . The angle between the asymptotic lines satisfies the sine Gordon equation. Bäcklund discovered the transformations which bear his name using line congruences. The Korteweg-de Vries equation dates back to 1895 as a description of shallow water waves. The more recent nonlinear Schrödinger equation describes a plethora of phenomena including the propagation of signals in fiber optic cables. Terng and I wrote an expository article for the Notices describing more of the history TU00. Chern was, of course, interested in the geometry described by the first of these equations. He wanted to extend Bäcklund transformations to affine minimal surfaces in . Chuu-Lian read Chern’s notes on affine geometry, and they obtained results very fast. However, they only obtained a single transformation without a parameter. After they obtained the results, the senior author said “You write it up” CT80. It is in stories like this that we get hints of how we might have been better thesis advisors and mentors.

3.2. Higher-dimensional versions

Chern then put Chuu-Lian in touch with Keti Tenenblat, a Turkish-Brazilian mathematician visiting Chern. The two women collaborated on the problem, suggested by Chern, of generalizing the classical Bäcklund transformations to hyperbolic -dimensional submanifolds in . Later Ablowitz, Beals and Tenenblat found a Lax pair and did the scattering and inverse scattering. Chuu-Lian found a loop algebra given by involutions which have a suitable splitting. There was a lot of interest in these equations at the time. I recall driving Chuu-Lian to the University of Chicago where she had been invited to talk on this work to the theoretical physics group.

3.3. Chinese women mathematians

Chuu-Lian Terng was not an isolated Chinese woman mathematician. As we noted before, students from the National Taiwan University routinely went on to PhD’s in the United States. The year before Chuu-Lian, a number of women whose names you may recognize graduated: Alice Chang, Fan Chung, Winnie Li, Gloria Wu. The success of this group of Chinese women mathematicians is not part of a general trend, and cannot be easily explained. Chuu-Lian did know about the other four, and discovered only recently that all of them have similar backgrounds. They all credit their mothers and an excellent education system which was open to women. She took her teachers’ questionings “Did your father help you with this?”, “Did your boyfriend help you study?” and “You don’t like to look up the answers in the back of the book?” as praise. As we attempt to diversify the pool of individuals succeeding in research mathematics, we might pay attention to what worked here. I recommend a forthcoming article by Allyn Jackson about this group of Chinese women mathematicians.

4. Midcareer

After her postdoc at Berkeley, Terng moved on to an assistant professorship at Princeton. Princeton had only started admitting women students in 1969, and admission had become gender blind in 1974. Chuu-Lian came to Princeton in 1978 as the first female assistant professor in the mathematics department. She was one of ten assistant professors who used their time at Princeton to get as much research done and move on to a “real” tenure track or tenured position elsewhere. Inconveniences, such as having an office on the tenth floor when the only women’s bathroom was on the third floor, happened all too often in those days. But in general, it was a matter of being ignored rather than treated badly.

When I came as a member to the Institute for Advanced Study for the academic year 1979–1980, we tried to work together for the first time. I recall that we had offices next to each other in Building C, and at least once Professor Langland slammed the door of the office we were working in. Our husbands have also noted that we are not quiet when we work together. Perhaps we were expected “to be seen but not heard”? I had been learning Teichmüller theory and was attending Bill Thurston’s course at the university. We learned about hyperbolic three manifolds which fiber over the circle topologically. No natural geometric fibration was known. With my background in minimal surfaces and Chuu-Lian’s command of tools in geometry, the problem seemed made for us.

Alas, it was not to be. The problem is still unsolved. When Bill Thurston heard of our project, he suggested that these manifolds might be fibered by minimal surfaces. We thought it more likely that there would be a fibration by constant mean curvature surfaces. In 1979 the problem was esoteric, but it is now known that every compact hyperbolic 3-fold has a cover that topologically fibers over a circle; these are now essential examples in the study of hyperbolic 3-folds.

I return to this problem periodically, but Chuu-Lian’s next project explored polar actions and isoparametric submanifolds. Due to the length limit of this article, we will refer the readers to a survey article by G. Thorbergsson [Th] and the book written by Terng and Palais PT88 on these topics. This research was carried out during Terng’s years at Northeastern University. These were not easy years with heavy teaching load, an unpleasant commute, and the kind of mistreatment and outright sexism that was all too common (which none of us enjoy revisiting). On the good side, she had good interactions with Boston area mathematicians and has fond memories of weekly joint seminar with her colleagues in topology, PDE, and geometry. She also benefited from lectures on integrable systems by Mark Adler and Pierre van Moerbeke at Brandeis. She moved from Northeastern University to University of California at Irvine in 2004 and enjoyed very much the differential geometry group, the graduate students, a house on campus, and the wonderful climate.

5. Soliton Equations in Geometry

My road into integrable systems was from the opposite direction from Chuu-Lian’s introduction via Chern. Through one of my PhD students Louis Crane, I learned about the interest of the physics community in “loop groups,” and read the papers of Louis Dolan on the sigma model. I had kept up with Chuu-Lian and her husband, only partly because I had relatives in the Boston area. After I noted that Dolan’s loop group actions were dressing actions and the classical Bäcklund transformations for sine Gordon were given by actions of some rational loops, we started to work together again. This section is intended to connect examples of integrable equations with equations familiar to geometers.

5.1. Finite dimensions

In finite dimensions, there is a definition of completely integrable.

Definition 5.1.1.

Let be a symplectic manifold. The Hamiltonian system given by is

where is the dual of with respect to , i.e.,

Definition 5.1.2.

is a conservation law for the Hamiltonian system given by if for every solution , , or equivalently,

A Hamiltonian system given by on a manifold of dimension is completely integrable if there are m conservation laws , which are in involution with each other, i.e., .

If two Hamiltonians are in involution with each other, their flows commute. This is the picture in finite dimensions. However, in infinite dimensions, the integrable systems in this article always have an infinite number of conservation laws whose flows commute. Only in very rare cases is there a theory which indicates “completeness.” The example I know involves finding action angle coordinates for a special case of KdV. Nevertheless, we shall see that the infinite-dimensional theory is much richer than the finite-dimensional theory.

The symplectic manifolds in this section are the Grassmanians , which are adjoint orbits in the Lie algebra of ,

Geometric information 5.1.3.

For ,

(i)

.

(ii)

.

(iii)

The orthogonal projection of on is

(iv)

The symplectic form is

(v)

The associated complex structure in the tangent space is

It is important to know at least one example.

Example 5.1.4.

Let and let . Then the hamiltonian flow for is

Note that if commute, then

However, when , , does not imply that . So the integrals for the Hamiltonian flow are not in involution. In this case, the symmetries are Poisson and we can expect to see such symmetries for equations with target with .

5.2. Equations of global analysis

We now switch gears. Geometers are familiar with the nonlinear elliptic equation of harmonic maps between Riemannian manifolds and associated parabolic heat and hyperbolic wave map equations. However, in the case that the image manifold is symplectic, we have a geometric nonlinear Schrödinger (GNLS) equation.

As a general principle, the image negatively curved symmetric spaces correspond to defocusing equations and those with positive curvature correspond to focusing. Formally they appear similar, but technically and geometrically they are very different. Terng and I considered the case of the positively curved symmetric space described above.

We briefly describe the GNLS on with image . We consider

where is the Schwartz space and . Corresponding to the geometry listed in 5.1.3, we have the following geometry in .

5.2.1. Geometric information in

For ,

(i)

.

(ii)

The formal inner product on the tangent space is

(iii)

The orthogonal projection of on is

(iv)

The symplectic structure on is

(v)

The associated complex structure in the tangent space to at is

Theorem 5.2.2.

The Hamiltonian flow for is the GNLS equation

We will explain in Section 6 why this equation is integrable for .

5.3. Ward harmonic maps and space-time monopoles

The 2- and -dimensional examples of harmonic maps and wave maps into are examples of equations which have many of the usual properties of soliton equations. However, there is a variant of the wave map equation in into due to Ward which has solitons and scattering and inverse scattering theories. We refer to Ward’s equation as the modified wave map. The wave map equation for is

Ward’s equation is

The Lax pair for this equation is more easily seen if we transform using the variables and . Then the modified wave map equation simplifies to

Proposition 5.3.1.

There is a Lax pair for the modified wave map of the form , where

The frame is useful for constructing a large number of examples, which are not known for the wave map. The modified wave map equation can be transferred (via a gauge transformation similar to the Hasimoto transformation) to the space-time monopole equation, which is a reduction of anti-self-dual Yang-Mills with signature to a monopole equation in .

Proposition 5.3.2 (DTU06).

Solutions to the Ward equation can be identified with special solutions to the space-time monopole equation in ,

Here is the curvature 2-form, is the (scalar) Higgs field in and is the map from 1-forms to 2-forms induced by the indefinite metric on .

The analytic properties of the wave map problem are hard theorems in geometric analysis and it is difficult to find explicit examples. Description of Euclidean monopoles are only accurate at large spacings. However there is both a scattering theory for the monopole equation and Bäcklund transformations for Ward’s equation which are gauge equivalent to the Lax pair for the monopole equation. This is due to the existence of Lax pairs for both variants of the equation which are gauge equivalent. For smooth decaying initial data, the scattering theory shows that solutions exist for all time. Terng’s papers DT07 and DTU06 contain a wealth of information about these solutions.

6. The Drinfel′d-Sokolov Construction for the MNLS Hierarchy

In this section, we outline a very general procedure developed by many mathematicians of different backgrounds for generating soliton equations and their properties. There is a long list of contributors to this project, starting with Zhakarov and Shabat in 1976. In the last and final section we will come back to Terng’s ideas for classifying them. Much of the work of Terng is based on a fundamental paper of Drinfel′d and Sokolov DdS84; hence I have called the general scheme a Drinfel′d-Sokolov (DS) construction. The flows are traditionally in the variables . However, certain pieces of the structure can be applied with only part of the structure available. For example, Bäcklund transformations can be applied when there is only a single flow if there is a Lax pair description. In perhaps most examples, the needed factorizations for scattering theory and Bäcklund transformations are only local. The matrix nonlinear Schrödinger equation (MNLS) is a generalization of the scalar NLS, and our choice is motivated by the fact that the factorizations are well-understood in this case. Also note that there may be multiple descriptions of the same flows. The computations do not need to be done in any fixed order.

6.1. Choice of a triple group

Most flows are based on a triple loop group . Here are subgroups of and elements of factor into products in both orders on an open, dense set. The loop parameter plays the role of a spectral parameter , and the group factoring is done via Birkhoff factorization. These groups dependent on a spectral or loop parameter are often called “loop groups,” especially in the physics literature. There exists an open dense set of (big cell) such that for , we can factor

In our example:

At the Lie algebra level we have . We can decompose a Laurent series converging in into negative powers of and nonnegative powers of . At the Lie group level, this is called Birkhoff factorization and involves writing a power series as the products of series that converge in the two regions. It fails on the group, but is valid on a dense open subset of what we call we call “the big cell.”

6.2. Choice of a vacuum frame

A vacuum frame determines the flows. Different flows result from different choices. For our example

with k +’s and minuses. Here the upper limit is arbitrarily large. We never use infinite sums.

6.3. The dependent variable in the flow

Our first step is completely formal. Choose an arbitrary element . Factor

for , .

Note that the scattering data does not depend on the flow variables . Since does, and do as well. We call the frame and the reduced frame (also called the Baker function). The frame has positive powers of and the reduced frame has negative powers:

Then (or technically ) will be our dependent variable in the flow, where is the coefficient of the power series of . Once the theory is sufficiently developed, plays the role of the dual of the element defining a central extension of . Note that is in and the flows generated by this framework are equations for maps .

If , is identically 0. This is the “vacuum” solution.

6.4. Lax pairs

Let

The dependent variable for the flow is . To get , we compare the coefficients of to get , or

the -th flow in the MNLS hierarchy.

It is not necessarily true that the are determined algebraically, which was first proved by Sattinger for our example by using the following two equations.

Lemma 6.4.1.

The are polynomials in and its derivatives in of order up to .

The Lax pair formulation is now

and . For , we call a frame for the MNLS hierarchy if is holomorphic for and or equivalently