Eugenio Calabi

1.1. Birth and Family

Eugenio Calabi was born in Milan, Italy, on May 11, 1923, into a Jewish family. His sister, Tullia Zevi, became a notable journalist.

1.2. Emigration due to Racial Laws

In 1938, his family was forced to leave Italy due to the Italian racial laws, subsequently arriving in the United States in 1939. This early experience of displacement significantly impacted his formative years.

1.3. Education

At the age of 16, in the fall of 1939, Calabi enrolled at the Massachusetts Institute of Technology (MIT) to study chemical engineering. His studies were interrupted in 1943 when he was drafted into the U.S. military and served during World War II. After his discharge in 1946, he completed his bachelor's degree under the G.I. Bill and was recognized as a Putnam Fellow. He earned a master's degree in mathematics from the University of Illinois Urbana-Champaign in 1947, and his PhD in mathematics from Princeton University in 1950. His doctoral dissertation, titled "Isometric complex analytic imbedding of Kähler manifolds", was supervised by Salomon Bochner.

2.1. Professorships and Affiliations

From 1951 to 1955, Eugenio Calabi served as an assistant professor at Louisiana State University. In 1955, he moved to the University of Minnesota, where he became a full professor in 1960. Calabi joined the mathematics faculty at the University of Pennsylvania in 1964. Following the retirement of Hans Rademacher, he was appointed to the prestigious Thomas A. Scott Professorship of Mathematics at the University of Pennsylvania in 1968. He assumed emeritus status in 1994, and in 2014, the university awarded him an honorary doctorate of science.

3.1. Major Honors and Fellowships

Beyond being a Putnam Fellow during his undergraduate studies, Calabi received several major honors. In 1982, he was elected to the National Academy of Sciences. He was awarded the Leroy P. Steele Prize from the American Mathematical Society in 1991, with the citation recognizing his "fundamental work on global differential geometry, especially complex differential geometry," which was noted for having "profoundly changed the landscape of the field." In 2012, he became a Fellow of the American Mathematical Society. For his contributions, he was awarded the rank of Commander of the Order of Merit of the Italian Republic in 2021.

4.1. Kähler Geometry

Calabi's foundational work in Kähler geometry is among his most celebrated contributions. At the 1954 International Congress of Mathematicians, he announced a theorem concerning the prescription of the Ricci curvature of a Kähler metric. Although his initial proof contained a flaw, the result became famously known as the Calabi conjecture. In 1957, Calabi published a paper where the conjecture was presented as a proposition with an openly incomplete proof. He provided a complete proof for the uniqueness of any solution, but the problem of existence was reduced to establishing a priori estimates for certain partial differential equations. In the 1970s, Shing-Tung Yau successfully proved the conjecture, leading to several striking consequences in algebraic geometry. A specific outcome of the conjecture's validity was the establishment of Kähler metrics with zero Ricci curvature on various complex manifolds, now known as Calabi-Yau metrics. These have become profoundly significant in string theory research since the 1980s.

In 1982, Calabi introduced a geometric flow, now called the Calabi flow, as a method for finding Kähler metrics with constant scalar curvature. More broadly, Calabi originated the concept of an extremal Kähler metric, demonstrating that these metrics provide strict global minima of the Calabi functional and that any constant scalar curvature metric is also a global minimum. Later, Calabi and Xiuxiong Chen conducted an extensive study of the metric introduced by Toshiki Mabuchi, showing that the Calabi flow contracts the Mabuchi distance between any two Kähler metrics. They further demonstrated that the Mabuchi metric endows the space of Kähler metrics with the structure of an Alexandrov space of nonpositive curvature, despite the technical challenges posed by the low differentiability of geodesics in this infinite-dimensional context.

Another notable construction by Calabi involves placing complete Kähler metrics on the total spaces of hermitian vector bundles whose curvature is bounded below. When the base is a complete Kähler-Einstein manifold and the vector bundle has rank one and constant curvature, this construction yields a complete Kähler-Einstein metric on the total space. In the specific case of the cotangent bundle of a complex space form, a hyperkähler metric is obtained. The Eguchi-Hanson space is a special instance of Calabi's general construction.

4.2. Geometric Analysis

Calabi made fundamental contributions to geometric analysis. He discovered the Laplacian comparison theorem in Riemannian geometry, which establishes a relationship between the Laplace-Beltrami operator (applied to the Riemannian distance function) and the Ricci curvature. Given that the Riemannian distance function is generally not differentiable everywhere, formulating a global version of this theorem presented a significant challenge. Calabi addressed this by employing a generalized notion of differential inequalities, which predated the later viscosity solutions introduced by Michael G. Crandall and Pierre-Louis Lions. By extending the strong maximum principle of Eberhard Hopf to his concept of viscosity solutions, Calabi was able to use his Laplacian comparison theorem to extend recent results by Joseph Keller and Robert Osserman to Riemannian contexts. Further extensions, based on different applications of the maximum principle, were later developed by Shiu-Yuen Cheng and Shing-Tung Yau, among others.

In parallel to the classical Bernstein problem for minimal surfaces, Calabi investigated the analogous problem for maximal surfaces, providing a solution in low dimensions. A comprehensive answer was later found by Cheng and Yau, who utilized the Calabi trick-a method pioneered by Calabi to overcome the non-differentiability of the Riemannian distance function. In related work, Calabi had earlier examined the convex solutions of the Monge-Ampère equation defined across all of Euclidean space with a 'right-hand side' equal to one. Konrad Jörgens had previously studied this problem for functions of two variables, proving that any solution is a quadratic polynomial. By interpreting the problem through the lens of affine geometry, Calabi was able to apply his earlier work on the Laplacian comparison theorem to extend Jörgens' findings to some higher dimensions. The problem was ultimately and completely resolved by Aleksei Pogorelov, and the combined result is commonly known as the Jörgens-Calabi-Pogorelov theorem.

Later, Calabi turned his attention to the problem of affine hyperspheres, initially characterizing such surfaces as those for which the Legendre transform solves a specific Monge-Ampère equation. By adapting his earlier methods used in extending Jörgens' theorem, Calabi successfully classified the complete affine elliptic hyperspheres. Further significant results in this area were subsequently obtained by Cheng and Yau.

4.3. Differential Geometry

Calabi's contributions to differential geometry extend beyond Kähler geometry and geometric analysis, encompassing various other areas. In 1953, Calabi and Beno Eckmann jointly discovered the Calabi-Eckmann manifold. This manifold is particularly notable as a simply-connected complex manifold that does not admit any Kähler metrics.

Inspired by recent work of Kunihiko Kodaira, Calabi and Edoardo Vesentini investigated the infinitesimal rigidity of compact holomorphic quotients of Cartan domains. Utilizing the Bochner technique and Kodaira's advancements in sheaf cohomology, they successfully proved the rigidity of higher-dimensional cases. Their work significantly influenced later research by George Mostow and Grigori Margulis, whose global rigidity results emerged from efforts to understand infinitesimal rigidity findings such as those by Calabi and Vesentini, along with related works by Atle Selberg and André Weil.

Calabi and Lawrence Markus explored the problem of space forms of positive curvature within Lorentzian geometry. Their findings, described as "very surprising" by Joseph A. Wolf, assert that the fundamental group must be finite. Furthermore, they showed that the corresponding group of isometries of de Sitter spacetime (under an orientability condition) will act faithfully by isometries on an equatorial sphere. As a result, their space form problem effectively reduces to the problem of Riemannian space forms of positive curvature.

In his PhD thesis, Calabi had previously considered the specialized case of holomorphic isometric embeddings into complex-geometric space forms, building upon John Nash's work in the 1950s on the flexibility of isometric embeddings. A striking result from Calabi's thesis demonstrated that such embeddings are entirely determined by the intrinsic geometry and the curvature of the space form in question. Moreover, he investigated the problem of existence through his introduction of the diastatic function, a locally defined function constructed from Kähler potentials that mimics the Riemannian distance function. Calabi proved that a holomorphic isometric embedding must preserve the diastatic function, which allowed him to derive a criterion for the local existence of holomorphic isometric embeddings. Later, Calabi studied two-dimensional minimal surfaces (of high codimension) in round spheres. He proved that the area of topologically spherical minimal surfaces can only assume a discrete set of values, and that the surfaces themselves are classified by rational curves within a specific hermitian symmetric space.

5.1. Family

Eugenio Calabi married Giuliana Segre in 1952. They had a son and a daughter.

6.1. Date and Circumstances

Eugenio Calabi died on September 25, 2023, at the age of 100, at his home in Bryn Mawr.

7.1. Impact on Mathematics and Physics

His work on the Calabi conjecture and the subsequent development of Calabi-Yau manifolds, particularly after Shing-Tung Yau's proof of the conjecture, proved to be foundational. These concepts became especially significant in string theory research starting in the 1980s, providing essential geometric frameworks for understanding the universe at its most fundamental level. His influence continues to be felt across various branches of mathematics and theoretical physics.

8.1. Selected Research Papers

• Calabi, Eugenio. "Isometric imbedding of complex manifolds." Annals of Mathematics, Second Series, volume 58, issue 1 (1953): pages 1-23. [https://doi.org/10.2307/1969817]
• Calabi, Eugenio; Eckmann, Beno. "A class of compact, complex manifolds which are not algebraic." Annals of Mathematics, Second Series, volume 58, issue 3 (1953): pages 494-500. [https://doi.org/10.2307/1969750]
• Calabi, E. "The space of Kähler metrics." Proceedings of the International Congress of Mathematicians, 1954. Volume II (1954): pages 206-207.
• Calabi, Eugenio. "On Kähler manifolds with vanishing canonical class." Algebraic Geometry and Topology, A symposium in honor of S. Lefschetz (1957): pages 78-89. [https://doi.org/10.1515/9781400879915-006]
• Calabi, E. "An extension of E. Hopf's maximum principle with an application to Riemannian geometry." Duke Mathematical Journal, volume 25 (1958): pages 45-56. [https://doi.org/10.1215/S0012-7094-58-02505-5]
• Calabi, Eugenio. "Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens." Michigan Mathematical Journal, volume 5, issue 2 (1958): pages 105-126. [https://doi.org/10.1307/mmj/1028998055]
• Calabi, Eugenio; Vesentini, Edoardo. "On compact, locally symmetric Kähler manifolds." Annals of Mathematics, Second Series, volume 71, issue 3 (1960): pages 472-507. [https://doi.org/10.2307/1969939]
• Calabi, E.; Markus, L. "Relativistic space forms." Annals of Mathematics, Second Series, volume 75, issue 1 (1962): pages 63-76. [https://doi.org/10.2307/1970419]
• Calabi, Eugenio. "Minimal immersions of surfaces in Euclidean spheres." Journal of Differential Geometry, volume 1, issue 1-2 (1967): pages 111-125. [https://doi.org/10.4310/jdg/1214427884]
• Calabi, Eugenio. "Examples of Bernstein problems for some nonlinear equations." Global Analysis, Proceedings of Symposia in Pure Mathematics, volume 15 (1970): pages 223-230. [https://doi.org/10.1090/pspum/015]
• Calabi, Eugenio. "Complete affine hyperspheres. I." Symposia Mathematica, volume X (1972): pages 19-38.
• Calabi, E. Métriques kählériennes et fibrés holomorphes_{Kähler metrics and holomorphic vector bundles}^French. Annales Scientifiques de l'École Normale Supérieure, Quatrième Série, volume 12, issue 2 (1979): pages 269-294. [https://doi.org/10.24033/asens.1367]
• Calabi, Eugenio. "Extremal Kähler metrics." Seminar on Differential Geometry, Annals of Mathematics Studies, volume 102 (1982): pages 259-290. [https://doi.org/10.1515/9781400881918-016]
• Calabi, Eugenio. "Extremal Kähler metrics. II." Differential Geometry and Complex Analysis (1985): pages 95-114. [https://doi.org/10.1007/978-3-642-69828-6_8]
• Calabi, E.; Chen, X. X. "The space of Kähler metrics. II." Journal of Differential Geometry, volume 61, issue 2 (2002): pages 173-193. [https://doi.org/10.4310/jdg/1090351383]

His collected works were published in 2021:

• Calabi, Eugenio. Collected Works. Edited by Jean-Pierre Bourguignon, Xiuxiong Chen, and Simon Donaldson. Springer, Berlin (2021).