Eugenio Calabi

Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications.

Calabi was born in Milan, Italy on May 11, 1923, into a Jewish family.

His sister was the journalist Tullia Zevi Calabi.

In 1938, the family left Italy because of the racial laws, and in 1939 arrived in the United States.

In the fall of 1939, aged only 16, Calabi enrolled at the Massachusetts Institute of Technology, studying chemical engineering.

His studies were interrupted when he was drafted in the US military in 1943 and served during World War II.

Upon his discharge in 1946, Calabi was able to finish his bachelor's degree under the G.I. Bill, and was a Putnam Fellow.

He received 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 done under the supervision of Salomon Bochner.

From 1951 to 1955 he was an assistant professor at Louisiana State University, and he moved to the University of Minnesota in 1955, where he became a full professor in 1960.

Calabi married Giuliana Segre in 1952, with whom he had a son and a daughter.

He turned 100 on May 11, 2023, and died on September 25.

Calabi made a number of contributions to the field of differential geometry.

Other contributions, not discussed here, include the construction of a holomorphic version of the long line with Maxwell Rosenlicht, a study of the moduli space of space forms, a characterization of when a metric can be found so that a given differential form is harmonic, and various works on affine geometry.

In the comments on his collected works in 2021, Calabi cited his article "Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens" as that which he was "most proud of".

At the 1954 International Congress of Mathematicians, Calabi announced a theorem on how the Ricci curvature of a Kähler metric could be prescribed.

He later found that his proof, via the method of continuity, was flawed, and the result became known as the Calabi conjecture.

In 1957, Calabi published a paper in which the conjecture was stated as a proposition, but with an openly incomplete proof.

He gave a complete proof that any solution of the problem must be uniquely defined, but was only able to reduce the problem of existence to the problem of establishing a priori estimates for certain partial differential equations.

In 1964, Calabi joined the mathematics faculty at the University of Pennsylvania.

Following the retirement of Hans Rademacher, he was appointed to the Thomas A. Scott Professorship of Mathematics at the University of Pennsylvania in 1968.

In the 1970s, Shing-Tung Yau began working on the Calabi conjecture, initially attempting to disprove it.

After several years of work, he found a proof of the conjecture, and was able to establish several striking algebro-geometric consequences of its validity.

As a particular case of the conjecture, Kähler metrics with zero Ricci curvature are established on a number of complex manifolds; these are now known as Calabi–Yau metrics.

They have become significant in string theory research since the 1980s.

In 1982, Calabi was elected to the National Academy of Sciences.

In 1982, Calabi introduced a geometric flow, now known as the Calabi flow, as a proposal for finding Kähler metrics of constant scalar curvature.

More broadly, Calabi introduced the notion of an extremal Kähler metric, and established (among other results) that they 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 made an extensive study of the metric introduced by Toshiki Mabuchi, and showed that the Calabi flow contracts the Mabuchi distance between any two Kähler metrics.

Furthermore, they showed that the Mabuchi metric endows the space of Kähler metrics with the structure of a Alexandrov space of nonpositive curvature.

The technical difficulty of their work is that geodesics in their infinite-dimensional context may have low differentiability.

A well-known construction of Calabi's puts complete Kähler metrics on the total spaces of hermitian vector bundles whose curvature is bounded below.

In the case that the base is a complete Kähler–Einstein manifold and the vector bundle has rank one and constant curvature, one obtains a complete Kähler–Einstein metric on the total space.

In the case of the cotangent bundle of a complex space form, one obtains a hyperkähler metric.

The Eguchi–Hanson space is a special case of Calabi's construction.

Calabi found the Laplacian comparison theorem in Riemannian geometry, which relates the Laplace–Beltrami operator, as applied to the Riemannian distance function, to the Ricci curvature.

He won the Leroy P. Steele Prize from the American Mathematical Society in 1991, where his "fundamental work on global differential geometry, especially complex differential geometry" was cited as having "profoundly changed the landscape of the field".

In 1994, Calabi assumed emeritus status, and in 2014 the university awarded him an honorary doctorate of science.

In 2012, he became a fellow of the American Mathematical Society.

In 2021, he was awarded Commander of the Order of Merit of the Italian Republic.