E. Calabi (Author), J.-P. Bourguignon, X. Chen, S. Donaldson (eds.): “Collected Works”

In May 2023, Eugenio Calabi, one of the most influential differential geometers of the 20th century, celebrated his 100th birthday. Springer honoured this calm giant by publishing his Collected Works, and we are greatly indebted to the three editors Jean-Pierre Bourguignon, Xiuxion Chen, and Simon Donaldson as well as the contributors to this volume for the tremendous work they invested to catch the fascinating mathematics— and personality—of Eugenio Calabi, most often just called ‘Gene’ Calabi. A few weeks before this article went into print, Gene Calabi died on September 25, 2023 after a fulfilled life as a mathematician, husband, father, and friend. I do not claim to be able to write an obituary, but with this article I would like to pay my personal tribute to a person who has shaped modern differential geometry during more than 50 years.

In modern times, people leave a trail of digital foot prints and some tend to be omnipresent, or at least it seems so. Eugenio Calabi is not one of them. He was an influencer before the letter with almost no digital presence; the only two exceptions the author is aware of are a one-hour interview conducted by Claude Lebrun for the Simons Foundation in 2019 [6] and a 10-minute talk delivered at a conference in honour of his 90th birthday in 2013 [4]. I recommend watching both; they definitely convey that Calabi had a very fine sense of humour. He stayed at the same university (the University of Pennsylvania) for most of his career (1964-1994), but was never a director of a prestigious research institute or held any positions in science administration—he clearly didn’t need this. A book is thus the right way to honour such a remarkable mathematician. The following contributions to Calabi’s life and work constitute the first part of the book, about 110 pages (some of them had been written earlier for Calabi’s seventieth birthday):

• 1. Shing-Tung Yau, An essay on Eugenio Calabi,

• 2. Blaine Lawson, Reflections on the early work of Eugenio Calabi,

• 3. Marcel Berger, Encounter with a geometer: Eugenio Calabi,

• 4. Jean-Pierre Bourguignon, Eugenio Calabi’s short biography and Eugenio Calabi and Kähler metrics,

• 5. Claude Lebrun, Eugenio Calabi and the curvature of Kähler manifolds,

• 6. Xiuxiong Chen and Simon Donaldson, Calabi’s work on affine differential geometry and results of Bernstein type.

They are a treasure of mathematical insight and personal anecdotes, providing a wider context to Calabi’s work. It goes without saying that this review is deeply influenced by all of the sources just mentioned, which I deeply enjoyed reading.

Eugenio Calabi was born in Milano, Italy, on May 11, 1923, into a Jewish family (Fig. 1 depicts him together with S.-S. Chern in Oberwolfach, 1976); his father Giuseppe was a lawyer. Under the impact of the Spanish Civil War (1936-1939), the family considered emigrating to the United States and quickly did so after the enactment of racial laws in 1938 (with short interims stays in Geneva and Paris), so that Eugenio Calabi arrived in his new adopted country of America in 1939 at the age of 16. He was immediatly accepted at MIT, though he then did not study mathematics, but rather chemical engineering. He became an American citizen in 1943 and was drafted to the US military the same year, and served for two years, mainly in Europe. After his return to MIT, he finished his bachelor and although he started looking into jobs in chemistry, had decided to switch to mathematics. He received a master’s degree in mathematics from the University of Illinois Urbana-Champaign in 1947.

Fig. 1

Eugenio Calabi (right) and Shiing-Shen Chern in Oberwolfach, 1976 (courtesy of Dirk Ferus). Photo ID: 7484. Autor: Dirk Ferus Quelle: Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach

By then, his parents had returned to Milano in 1946, since his father could not practice in the US. Of his three siblings, his sister Tullia (1919-2011), who had finished her studies in philosophy and music in the United States and became a political journalist, deserves special mention. She also returned to Italy in 1946 with her husband, the architect Bruno Zevi, whom she had married in 1940. She was active in the reconstruction of Jewish life in Italy, covered the Nuremberg and Eichmann trials as a correspondent, and was president of the UCEI, the Union of Italian Jewish Communities, from 1983 to 1988. In 1992, the then President of Italy, Oscar Luigi Scalfaro, awarded her the Grand Cross of the Order of Merit of the Italian Republic.

Let us return to Eugenio Calabi himself. After his master’s degree, he applied to Harvard and Princeton in 1947 and was admitted to both, but Princeton offered temporary housing, so he preferred Princeton. He completed his PhD thesis under the supervision of Salomon Bochner in 1950, entitled “Isometric Complex Analytic Imbedding of Kähler Manifolds”. As beautifully explained by Blaine Lawson in his contribution, the main question was: Which complex Hermitian manifolds $(X,h)$ can be holomorphically and isometrically embedded into ${\mathbb{C}}^{N}$ (or more generally into a complex space form)? In contrast to the Riemannian case, Calabi proved that such an embedding will be unique up to isometry, and that the existence of the embedding can be characterised by a certain intrinsic, pointwise condition. It makes use of a refinement of the Kähler potential, Calabi’s so-called diastatic function. The particular case of a holomorphic curve then leads to applications to the theory of minimal surfaces, both into Euclidian spaces and into spheres. In the interview [6], Calabi says about this work: “[…] it was the geometrical aspects that attracted me, and Kähler manifolds seemed to be the area to work in, for there was an inner structure that was really visible to me.” Already here, Calabi’s lifelong interest for the geometry of Kähler manifolds as objects not only of complex, but of differential geometry, becomes apparent! Finally, after graduating, he married Giuliana Segre in 1952, with whom he had a son and a daughter.

One of Calabi’s next papers, A class of compact, complex manifolds wich are not algebraic (with Beno Eckmann, 1953), turned out to be a game changer in complex geometry, despite being only 6 pages long. Almost nothing was known about complex non Kähler manifolds at the time, and Kodaira’s classification of compact complex analytic surfaces would only be published in 1963. Calabi and Eckmann showed that there exist complex structures on the products of two odd-dimensional spheres $M_{p,q}=S^{2p+1}\times S^{2q+1}, \ p,q\geq 1$, yet they cannot carry a Kähler metric for topological reasons. This was the starting point of the highly active field of complex non-Kähler geometry, to which generations of researchers have contributed since then. The classification of compact complex manifolds remains a tremendous challenge to this day.

One advantage of Collected Works is that they usually contain reprints of articles that may not easily be available in digital form. Here, this applies in particular to the summary of the 15-minutes talk entitled The space of Kähler metrics Calabi presented at the ICM 1954 in Amsterdam [2], see Fig. 2 (As it happens, his co-author, Beno Eckmann, also gave a talk, but not on their joint work, but on cohomology theory instead). On a closed (i.e., compact without boundary) Kähler manifold¹$(M^{2n},g,J)$, the Ricci curvature $\mathrm{Ric}$ defines a 2-form $\varrho$ of type $(1,1)$ through $\varrho (X,Y) = {\mathrm {Ric}}(JX,Y)$, and this 2-form turns out to be closed, hence it may be used as a representative of the Chern class of $M^{2n}$, i.e. the identity $[\varrho ] = 2\pi c_{1}(M^{2n})$ holds in $H^{2}(M^{2n},\mathbb{Z})$. It is thus a natural question to ask the converse question: On a closed Kähler manifold $(M^{2n},g,J)$, can any closed form of type $(1,1)$ whose cohomology class lies in $2\pi c_{1}(M^{2n})$ be the Ricci form of some unique (different) Kähler metric? Calabi had thought to have proven a positive answer, and this is what Theorem 1 of his abstract claims. He also outlined a method of proof that was later coined as the continuity method. In this approach, the problem may be restated as the question whether a non-linear, second order PDE of Monge-Ampère type admits a unique solution. Calabi showed uniqueness, but admitted in 1957 that his proof of the existence of a solution was seriously flawed. The problem then became known as the Calabi conjecture, which would remain open for almost 20 years! Thierry Aubin gave a partial solution for Kähler manifolds with positive bisectional curvature in 1970, and in 1976 Shing-Tung Yau gave a groundbreaking proof of the full conjecture which earned him the Fields Medal in 1982. Philosophically, the continuity method proved to be a successful strategy, but with seemingly insurmountable analytical challenges (a priori estimates is the right keyword for the expert in this context). The consequences of this theorem for mathematics (and, in modern times, for the more speculative branches of theoretical physics) are enormous. For example, in the special situation of vanishing first Chern class, the solution of the Calabi conjecture implies the existence of a Ricci-flat metric (although one cannot write it down explicitly)—by now, we call such manifolds Calabi-Yau manifolds. Ever since they were discovered, Calabi-Yau manifolds have been leading an exciting life on the borderline between differential geometry, algebraic geometry, and theoretical physics.

Fig. 2

The beginning of a long story: The first lines of Calabi’s abstract for his talk at the ICM 1954 in Amsterdam [2], stating as a theorem what would be named the Calabi conjecture later

But it would be a mistake to reduce Calabi’s work and broad geometric talent to the subject of Kähler metrics with prescribed Ricci curvature. It is much less well known that Calabi also made valuable contributions to affine geometry over the years. At some point in his career he bought a classic of affine geometry by Blaschke from the widow of a deceased colleague who had put her husband’s book up for sale; as he said in the interviews [5, 6], he was fascinated by it right away. Roughly speaking, affine geometry deals with the study of locally convex hypersurfaces in $\mathbb{R}^{n}$ that are invariant under the group of unimodular affine transformations. In a series of papers, Calabi described affine spheres, making clever use again of some adapted Monge-Ampère type equation. Moreover, he recognized that the methods and results had repercussions to others areas of geometry, like spacelike submanifolds of Lorentz manifolds or extremal metrics. Calabi was also interested in symplectic geometry. He only wrote one article on it [3] (again, it is hard to get because it appeared in a book in honour of his PhD advisor Salomon Bochner), but what an article! In it, he described the Lie algebra of symplectic vector fields and he defined the Calabi morphism on the space of Hamiltonian vector fields which developed into a crucial tool for the investigation of the group of volume-preserving homeomorphisms of a symplectic manifold.

Looking at his professional life, Calabi joined Louisiana State University from 1951 to 1955 as an assistant professor, then he moved on to the University of Minnesota. In 1964, Calabi was appointed professor at the University of Pennsylvania, where he remained until his retirement in 1994. Without knowing any details (these things rarely make it into the records), one can fairly assume that other departments may have made him attractive offers. Calabi has been honored with numerous awards and distinguished memberships, amongst which we cite: election to the National Academy of Sciences (1982), the Leroy P. Steele Prize of the American Mathematical Society (1991), Commander of the Order of Merit of the Italian Republic (2021).

According to the Math Genealogy project, Eugenio Calabi had only 5 students, of whom only one, Xiuxiong Chen, stayed in academia (he graduated from the University of Pennsylvania in 1994 and he is now a professor at Stony Brook and at Shanghai Tech, and director of the Centre for Geometry and Physics at the USTC in Hefei / China) and is in fact one of the editors of this remarkable book (the other four students are: Carlos Ferraris, Salvador Gigena, Thomas Ho, Jiangfan Li). According to himself [6], Calabi had a “lousy reputation as a lecturer”, but this is hard to imagine when reading testimonials of people who enjoyed mathematical discussions with him. In the collection of Memorable encounters with Calabi published in the EMS Magazine [1], Blaine Lawson writes: “Over time, I realized how generous he has always been to young mathematicians.” In 1982, Calabi had introduced a functional, now known as the Calabi functional, as a proposal for finding Kähler metrics of constant scalar curvature; its extremal points are called extremal Kähler metrics, a very active area of Hermitian geometry. In their joint work, Calabi and his last PhD student Chen made an extensive study of the space of all Kähler metrics on a given closed complex manifold; in particular, they proved that it is endowed with the structure of an Alexandrov space of nonpositive curvature. Xiuxiong Chen summarised the style of his supervision as follows [1]: “Prof. Calabi would pen his explanations or thoughts on whatever was available in hand or at hand, be it an envelope or a napkin, or a blackboard in a nearby classroom. We would talk hours and end often in his office, but also in the mail room and in the hallways. I would take home those envelopes and napkins (regrettably many of them got lost during our many moves), but most of the time I would jot down on my notepad what he wrote or said, or occasionally my own musings.”

There is a kind of books that has become rare nowadays—hard covered in dark linen, embossed in golden letters, with a matching sorting belt. I am glad that Springer decided to produce their series of Collected Works of seminal mathematicians in this style, and I hope that many libraries will purchase a copy in order to ensure that such high quality books will also be produced in the future. A physical book, and in particular such a beautiful one of over 800 pages, is a permanent invitation to flip through, to start reading somewhere, and start again at another randomly chosen page the next day. The more I did this, the more I got intimidated by the thought of writing a review on this mathematical marvel. Some articles are classics, while others touch upon subjects I wasn’t aware that Calabi had worked on them. All them were a pleasure to read, and so I invite you all to share the mathematical joy I have felt and to use the opportunity to look “over the shoulder” of Eugenio Calabi while creating new geometry.