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In July 2019, on behalf of the Simons Center for Geometry and Physics, Claude LeBrun, Stony Brook Mathematics Department, conducted an interview with Eugenio Calabi at the University of Pennsylvania Mathematics Department. Calabi is of course primarily famous for his pioneering work on Calabi-Yau manifolds, a class of solutions of Einstein’s equations that plays a central role in current research, both in mathematics (where they connect differential geometry and algebraic geometry in a remarkable new way) and in physics (where they are used to construct model universes via string compactification). This film is partly intended as a celebration of the recent award of the Veblen Prize to Simons Center and Stony Brook faculty members Simon Donaldson, Xiuxiong Chen, and Song Sun, for work answering questions first raised by Calabi. But, more specifically, it will also help us remember Calabi’s personal role in the story, both as Xiuxiong Chen’s thesis advisor, and as the mathematical grandfather of Chen’s student Song Sun.
CL: To begin, Gene, before we get to mathematics, could you say a little bit about your childhood and how you came to the United States?
EC: I grew up in Milan, Italy. My father was a lawyer. And he realized early on that I was gifted in mathematics. I remember when I was in first or second grade trying to memorize the multiplication table. My father taught me what a prime number was, and he told me, “your job is to find the law—how prime numbers succeed one another.” And it was a standard question to me: “have you found the law of prime numbers?”
I went to public school in Milan, through what you might call tenth grade. I left when I was about 15, with my family, and we spent a year in France waiting for our American visas. This was 1938-1939, just before World War II broke out.
CL: So you had already been through the horror of Mussolini’s racial laws and so forth?
EC: Yes. It had just begun in 1938. My father had been getting alarmed and he started planning a possible exit from Italy around 1936, at the end of the Ethiopian war and the breakout of the Spanish Civil War, in which Italy and Hitler were allies. And when the racial laws came out he decided there and then, overnight, that he had to put the family safe. And, so, we eventually came here in the spring of 1939.
CL: Well done. And how old were you when you got to the United States?
EC: Sixteen. And I got admitted to MIT right away. It was a various “quirk” of scheduling—I had skipped a year both entering and another one exiting France. I majored in chemical engineering at MIT, but by the time I graduated I had decided to switch to mathematics. I applied for graduate school at both Harvard and Princeton in 1947. I was admitted to both, but at Princeton they offered temporary housing for graduate students so I went there.
CL: (Laughs) Important decisions are often made on the flimsiest basis, right?
EC: Oh yes.
CL: But you do have to have a place to live.
EC: I also spent two terms at the University of Illinois. That was my real start in pure mathematics. In 1947 for the winter and spring terms. Then that summer I moved to Princeton.
CL: Did you start working with Salomon Bochner immediately?
EC: About a year in—1948.
CL: So Bochner was the person that knew something about differential geometry. In particular, he worked on Ricci curvature and he proved an important theorem about harmonic forms. He seemed to be very skeptical in that article about the Hodge–de Rham theorem. He talks about harmonic forms but he doesn’t talk about how they’re related to topology at all.
EC: That I learned a little later. I was strictly studying differential geometry and the problem of selecting metrics—the manifolds of many possible metrics—and whether there are any problems about the function space of all metrics. Nonlinear analysis was still at its very infancy at the time. But it was the geometrical aspects that attracted me, and Kähler manifolds seemed to be the area to work in.
CL: I guess Bochner also started working on Kähler geometry not very long after his paper on harmonic one forms. But he always refers to them as “so-called Kähler metrics.” He apparently knew that Kähler had been a Nazi and was very unhappy with the man over this.
EC: Well, I hadn’t noticed that. I had met Kähler when I just finished my doctorate in 1950, at the first International Congress of Mathematicians. I spoke with him briefly. My thesis was on Kähler metrics—the embedding problem. And I invented the word ‘diastasis’ in my thesis: the distance function. The normalized potential for Kähler metrics that works in the analytic case only.
CL: In 1954, you spoke at the next International Congress in Amsterdam, I believe? And that was when you first announced your entire program about representing Kähler classes on compact Kähler manifolds?
EC: Yes. I also wrote the first paper on the so-called generalization—on the theorem on the hessians, on real numbers—complex functions. At my first job at Louisiana State University, I bought a book on affine differential geometry, by Blaschke. And I was fascinated by it.
CL: So this work on real the Monge-Ampere equation—was this before or after you started thinking about the complex Monge-Ampere and Ricci flat Kähler metrics?
EC: Simultaneous. It was a step toward it. I was motivated by the other one but I also became interested in Affine differential geometry on its own right. And in fact, my current interest as it is—or at least pretends to be, at my age I can’t do much anymore—is still affine geometry.
CL: Still it’s quite amazing that you’re still doing mathematics. I saw you gave a very nice talk a couple of years ago. So how old are you now?
CL: 96. And still at it. That’s wonderful.
EC: Well, now it’s what we call research simulation.
CL: (Laughs) Well doing mathematics is such a kick, how could you possibly want to give it up?
EC: My favorite hobby.
CL: We’re very lucky that someone actually wants to pay us to pursue our hobbies.
EC: To follow your hobbies as a profession is the extraordinary luck I’ve had in my life.
CL: Yes. So, we were talking about some things that happened in 1954. I think that was the year you also submitted your paper for the Lefschetz Festschrift Volume, and that’s the one that lays out the Calabi conjecture. At the time, I think you thought at first maybe you could prove it, but then by the end of the paper you’re expressing grave doubts about one aspect of the continuity method.
EC: I remember hearing for the first time in my life, in that period, a priori estimates.
CL: (Laughs) Yes, so you wrote me a letter once in which you claimed that you had sent your paper for the Lefschetz volume to André Weil…
EC: Ah yes. And it was at a meeting in Trieste, Italy where I actually met Nirenberg and Bers. And I was talking to them about this problem and finally they told me you cannot solve partial differential equations without a priori estimates. And I said “what’s that?”
CL: (Laughs) I think this is a great tale for any young mathematician to hear because we all start off—even very, very talented people—start off from a position of relative ignorance, right? And then you have to talk to the experts and find out.
EC: The point is that you don’t learn during the course, you learn after, when mulling over the course. That has been my experience at least.
CL: Your paper on the Calabi conjecture is an amazing paper. It’s very, very impressive. But apparently you sent this to André Weil right? You must have been thinking that he was the expert on K3 surfaces?
EC: Well I asked him whether it was of any interest because I knew very little about algebraic geometry at the time. And he answered me very quickly and he said “how do you prove it?”
CL: He was a very brilliant man but not known for generosity. He was a very smart guy who liked to show that he was the smartest guy in the room.
EC: Oh yes. I was intimidated by him, actually.
CL: So you were Bochner’s student. And you had just given a talk at the Lefschetz Festschrift. This makes me wonder… I’ve heard all of these stories about how Bochner basically would not talk to Lefschetz. I’ve been told in particular Bochner did not want to be in the same room as Lefschetz.
EC: That’s right. When we had invited speakers, Lefschetz liked to get the students acquainted with the speakers, so he encouraged them to have a reception for the speakers at the graduate college. And the usual routine was that Lefschetz would come very early. And leave early. And then we’d call Bochner.
CL: So it’s a mystery that they were both very major mathematicians.
EC: They respected each other. I mean, at least I’ve heard Bochner quoting Lefschetz. The other way around, I did not follow Lefschetz too closely, but I think it happened.
CL: There’s certainly many results proved by later mathematicians that use ideas from Lefschetz combined with ideas from Bochner. So, for the benefit of our audience, it might be good to review just a little bit about the statement of what’s now called the Calabi-Yau theorem. That if you have a compact Kähler manifold then if you specify any volume form, provided it has the right integral, that you can represent, in a given Kähler class, you could find a unique metric with that volume form. And then this has the consequence that if the first Chern class of the Kähler manifold is zero, you can prove that there is a unique Ricci flat Kähler metric in each Kähler class. So, in your paper in the Lefschetz volume you give a truly beautiful geometric proof of the fact that the solution, if it exists, is unique. And this is just a beautiful argument. Basically, a maximum principle argument.
EC: Sure, it’s the Bochner principle.
CL: I definitely think that your paper is still worth reading because the uniqueness is so beautifully explained there. And of course, as it turned out, in terms of proving existence, you had one of the key ideas of the continuity method. But in terms of actually showing that the set of parameters for which you have a solution of the perturbed equation is closed, you needed other ideas and that’s only solved by Yau in the 1970s.
EC: That’s right. That’s another dramatic episode because when he first announced it, he had earlier thought he had the counterexample. And that was wrong. He recognized it but then he announced the proof shortly after, maybe less than a year. There was a great deal of excitement. We had to hear the details, so we had arranged a meeting as soon as we could. The meeting took place on Christmas day in Nirenberg’s office in New York. We met there in his office and heard the first proof. Which I did not understand.
CL: So there’s a C0 estimate which is done by iterating Lp spaces. But there’s also sort of a key C3 estimate that’s based on your earlier work in affine geometry. Did he discuss that in the same lecture?
EC: Yes he quotes it.
CL: You told me once that that was the first paper where you proved something with a priori estimates, right?
EC: That’s right. It was a celebration of the first understanding of analysis.
CL: So, with respect to your work on extremal Kähler metrics…Well actually one odd historical thing that some people even in the field may not know… There have been a lot of papers on Kähler-Einstein metrics with nonzero scalar curvature. And they have often put in the title that this is a proof of Calabi’s conjecture. Whereas you didn’t actually explicitly say that, although it follows from your assertions about constant scalar curvature.
EC: Oh yes. Well I realized only later that constant scalar curvature is obstructed in some manifolds.
CL: When did you first realize that?
EC: When the paper by Futaki appeared.
CL: Oh I see. So, you’d had these concerns about holomorphic vector fields – It’s there in your first article about constant scalar curvature Kähler metrics. You assume from the beginning that you’re on a manifold which supports no holomorphic vector fields, but you don’t say why that’s an important fact.
EC: Oh yes it was of interest to me as a variational problem. And then the first examples of extremal metrics in Ricci curvature is not constant. I even wrote a short paper announcing it.
CL: It’s a fairly long paper that’s in Yau’s conference volume. And there was a seminar on differential geometry and it’s a very beautiful article where you discuss the general variational problem, show what the Lagrange equations are, and then produce solutions on blow-ups. For example, CP2 blown up at a point supports solutions in every Kähler class, which have non-constant scalar curvature. When did you actually find those solutions? Was that just in the process of writing that paper?
EC: Yes. In the process, in the early 1980s.
CL: So those are now usually called extremal Kähler metrics, although it was not shown that they were minima for a long time. You had given a beautiful proof in the constant scalar curvature case that actually if you have a solution it is the minimum of the functional. It was, I think, your student Xiuxiong Chen that first proved that they were always minimizers in the general case.
CL: I’m just thinking back. You really made great strides in Kähler geometry as a branch of Riemannian geometry in the ’50s, ’60s, ’70s and ’80s. But back in the 1950s how well did people really even understand what a Kähler metric was?
EC: Well I understood it from the Bochner lectures. We asked him to give a course in differential geometry for the first time in my second year of graduate school, 1948-1949. And he defined it there.
CL: In particular it turns out that Kähler geometry is an example of Riemannian geometry of special holonomy. And that fact seems to have just been ignored by many people on the kind of algebro-geometric end of things.
EC: Well yes, it was really a remarkable discovery but it took years to focus from there—from its first announcement by Kähler.
CL: My colleagues Xiuxiong Chen and Simon Donaldson were kind enough to provide some questions that I should ask. One interesting question that Simon raises is that in some of your early papers you were interested in non-Kähler complex manifolds, in particular with Eckmann. You found a beautiful set of examples of complex structures on products of spheres which you know are on manifolds that certainly can’t admit Kähler metrics because their second cohomology is trivial. Have you followed these areas of non-Kähler complex geometry at all?
EC: No, just those questions that are on my mind, if there are any more examples other than the usual. Perhaps one being, after Milnor’s discoveries, whether you could also have twisted spheres?
CL: Well questions about complex structures on the sixth sphere are one of those annoying things that just won’t go away. It’s unfortunately one of those problems that is basically still quite out of range of any technology we have, and may always be.
EC: Yes, I’ve been bothered by that. But the obstruction to complex structures is basically unknown. Unless you have an almost complex structure. Obstruction to integrability.
CL: Another very interesting general question that Simon Donaldson proposed was, “when did you see analysis starting to have a major impact in differential geometry?”
EC: From the correspondence I had with André Weil.
CL: That’s when you realized that it was an essential thing?
EC: There was a problem. It was fundamentally analysis. I think that mathematicians very often what they do at a later age was originally in their mind much earlier.
CL: Maybe with 20/20 hindsight though sometimes you know your ideas are so confused when you’re young.
EC: Well I know I’m certainly confused when learning things. Learning is a confusing process and the real learning takes place after you’ve sort of digested the information. Learning is a digestive process, in a creative way.
CL: So in your own case—we mentioned before this article on extremal Kähler metrics that came out in the 1980s. In a way, you’re writing about things that you had started to think about years before but actually being forced to sit down and write out the details you’ve discovered new things.
EC: That’s right. Well that’s the digestive aspect.
CL: So, in the course of your career you’ve seen these enormous changes in the way that mathematics is communicated. Looking back it seems like the number of mathematicians, particularly mathematicians that did something that would be relevant to your own area, would have been relatively small in fact, where you would have probably known almost everybody.
EC: I used to.
CL: So I mean mathematics was usually disseminated by mail and telephone. And in particular, anything that was in the direction of a collaboration, or dependent upon, had to be done in those ways. But also, you could only exchange ideas with people if you already knew them.
EC: That’s why we went to meetings. Meetings were useful. I think they still are.
CL: Some of your early work was on complex manifolds and you were asking questions that were alien to algebraic geometers. Were there ever any meetings where you would meet people that were interested in these topics?
EC: Very few. The only one I knew was Bochner and his other students.
CL: So when you collaborated with Eckmann. How did you meet Eckmann?
EC: Heinz Hopf was visiting Princeton, and I told him about the construction. And Bochner balled me out for doing so. But Hopf told me that Eckmann had just done the same thing. And so, he put us in touch. I met him later, shortly after. We decided to write jointly by correspondence.
CL: When you look back, was there a certain period of time when you really thought that what you were doing was particularly fun and exciting?
EC: No. I was just finding my way.
CL: I’m just wondering though, you’re now you at a point in your career when your name is famous. People probably don’t realize Calabi is a person. Calabi-Yau is a phenomenon. You were telling me earlier that there was a dance performance called Calabi-Yau in New York. And so, it’s kind of entered in popular culture. Do you ever have the experience that people want to come up and ask you about physics or something like that, or string theory?
EC: No. I’ve heard about them, of course. But no, my favorite slogan to explain mathematics to the layman is “’it’s quintessentially science fiction.” I never quite understood the implications. But it was a piece of luck—unexpected.
CL: Well but on the other hand sometimes when you just do what comes naturally in mathematics it pays off, right?
EC: Yes. That’s my luck.
CL: Well, Gene it’s been wonderful interviewing you, I’m so glad that we could do this.
EC: Well, as I mentioned earlier it’s an ego trip for me.
CL: It’s a well-deserved one. I think that many people will find this interview interesting and I’m glad we were able to do it. Thanks a lot