Eugenio Calabi (born 11 May 1923) is an Italian-born American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and the
Thomas A. Scott Professor of Mathematics, Emeritus, at the
University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
, specializing in
differential geometry,
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
and their applications.
Academic career
Calabi was a
Putnam Fellow
The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regar ...
as an undergraduate at the
Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
in 1946.
He received his
PhD in mathematics from
Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
in 1950 after completing a doctoral dissertation, titled "Isometric complex analytic imbedding of
Kahler manifolds", under the supervision of
Salomon Bochner
Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry.
Life
He was born into a Jewish family in Podgórze (near Kraków), then Aus ...
. He later obtained a professorship at the
University of Minnesota
The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public land-grant research university in the Twin Cities of Minneapolis and Saint Paul, Minnesota, United States. ...
.
In 1964, Calabi joined the mathematics faculty at the
University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...
. Following the retirement of the German-born American mathematician
Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892, Wandsbeck, now Hamburg-Wandsbek – 7 February 1969, Haverford, Pennsylvania, USA) was a German-born American mathematician, known for work in mathematical analysis and number theory.
Biography
Rademacher r ...
, he was appointed to the
Thomas A. Scott Professorship of Mathematics at the University of Pennsylvania in 1967. He won the
Steele Prize from the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
in 1991 for his work in differential geometry. In 1994, Calabi assumed emeritus status. In 2012 he became a fellow of the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
. In 2021, he was awarded Commander of the
Order of Merit of the Italian Republic
The Order of Merit of the Italian Republic ( it, Ordine al Merito della Repubblica Italiana) is the senior Italian order of merit. It was established in 1951 by the second President of the Italian Republic, Luigi Einaudi.
The highest-ranking ...
.
Research
Calabi has made a number of fundamental contributions to the field of
differential geometry. Other contributions, not discussed here, include the construction of a holomorphic version of the
long line Long line or longline may refer to:
*'' Long Line'', an album by Peter Wolf
* Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fi ...
with
Maxwell Rosenlicht, a study of the
moduli space of
space form
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
s, a characterization of when a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
can be found so that a given
differential form is harmonic, and various works on
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...
. 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 is "most proud of".
Kähler geometry
At the 1954
International Congress of Mathematicians, Calabi announced a theorem on how the
Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
of a
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
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 the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
. 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 the 1970s,
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
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 introduced a
geometric flow, now known as the
Calabi flow, as a proposal for finding Kähler metrics of constant
scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
. 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
Toshiki Mabuchi (kanji: 満渕俊樹, hiragana: マブチ トシキ, Mabuchi Toshiki, born in 1950) is a Japanese mathematician, specializing in complex differential geometry and algebraic geometry. In 2006 in Madrid he was an invited speaker at th ...
, 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 In geometry, Alexandrov spaces with curvature ≥ ''k'' form a generalization of Riemannian manifolds with sectional curvature ≥ ''k'', where ''k'' is some real number. By definition, these spaces are locally compact complete length spaces where t ...
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.
Geometric analysis
Calabi found the ''Laplacian comparison theorem'' in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
, which relates the
Laplace–Beltrami operator, as applied to the
Riemannian distance function, to the Ricci curvature. The Riemannian distance function is generally not differentiable everywhere, which poses a difficulty in formulating a global version of the theorem. Calabi made use of a generalized notion of differential inequalities, predating the later
viscosity solution In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE) ...
s introduced by
Michael Crandall and
Pierre-Louis Lions
Pierre-Louis Lions (; born 11 August 1956) is a French people, French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Me ...
. By extending the
strong maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
of
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who ...
to his notion of viscosity solutions, Calabi was able to use his Laplacian comparison theorem to extend recent results of
Joseph Keller
Joseph Bishop Keller (July 31, 1923 – September 7, 2016) was an American mathematician who specialized in applied mathematics. He was best known for his work on the "geometrical theory of diffraction" (GTD).
Early life and education
Born i ...
and
Robert Osserman
Robert "Bob" Osserman (December 19, 1926 – November 30, 2011) was an American mathematician who worked in geometry. He is specially remembered for his work on the theory of minimal surfaces.
Raised in Bronx, he went to Bronx High School of ...
to Riemannian contexts. Further extensions, based on different uses of the
maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
, were later found by
Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and Yau, among others.
In parallel to the classical
Bernstein problem
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero m ...
for
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s, Calabi considered the analogous problem for
maximal surface In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold , a maximal surface is a spacelike submanifold of whose mean curvature is zero. ...
s, settling the question in low dimensions. An unconditional answer was found later by Cheng and Yau, making use of the ''Calabi trick'' that Calabi had pioneered to circumvent the non-differentiability of the Riemannian distance function. In analogous work, Calabi had earlier considered the convex solutions of the
Monge–Ampère equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is li ...
which are defined on all of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
and with 'right-hand side' equal to one.
Konrad Jörgens had earlier studied this problem for functions of two variables, proving that any solution is a quadratic polynomial. By interpreting the problem as one of
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...
, Calabi was able to apply his earlier work on the Laplacian comparison theorem to extend Jörgens' work to some higher dimensions. The problem was completely resolved later by
Aleksei Pogorelov
Aleksei Vasil'evich Pogorelov (russian: Алексе́й Васи́льевич Погоре́лов, ua, Олексі́й Васи́льович Погорє́лов; March 2, 1919 – December 17, 2002), was a Soviet and Ukrainian ...
, and the result is commonly known as the ''Jörgens–Calabi–Pogorelov theorem''.
Later, Calabi considered the problem of
affine hyperspheres, first characterizing such surfaces as those for which the
Legendre transform solves a certain Monge–Ampère equation. By adapting his earlier methods in extending Jörgens' theorem, Calabi was able to classify the complete affine elliptic hyperspheres. Further results were later obtained by Cheng and Yau.
Differential geometry
Calabi and
Beno Eckmann
Beno Eckmann (31 March 1917 – 25 November 2008) was a Swiss mathematician who made contributions to algebraic topology, homological algebra, group theory, and differential geometry.
Life
Born in Bern, Eckmann received his master's degree from ...
discovered the
Calabi–Eckmann manifold in 1953. It is notable as a
simply-connected complex manifold which does not admit any
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
s.
Inspired by recent work of
Kunihiko Kodaira
was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese ...
, Calabi and
Edoardo Vesentini
Edoardo Vesentini (31 May 1928 – 28 March 2020) was an Italian mathematician and politician who introduced the Andreotti–Vesentini theorem. He was awarded the Caccioppoli Prize in 1962.
Vasentini was born in Rome
, established_title ...
considered the infinitesimal rigidity of compact holomorphic quotients of
Cartan domains. Making use of the
Bochner technique and Kodaira's developments of
sheaf cohomology, they proved the rigidity of higher-dimensional cases. Their work was a major influence on the later work of
George Mostow
George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy o ...
and
Grigori Margulis, who established their renowned global rigidity results out of attempts to understand infinitesimal rigidity results such as Calabi and Vesentini's, along with related works by
Atle Selberg
Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarde ...
and
André Weil.
Calabi and Lawrence Markus considered the problem of
space form
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
s of positive curvature in
Lorentzian geometry. Their results, which
Joseph Wolf
Joseph Wolf (22 January 1820 – 20 April 1899) was a German artist who specialized in natural history illustration. He moved to the British Museum in 1848 and became the preferred illustrator for explorers and naturalists including David Livi ...
considered "very surprising", assert that the
fundamental group must be finite, and that the corresponding group of isometries of
de Sitter space
In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
time (under an orientability condition) will act faithfully by isometries on an equatorial sphere. As such, their space form problem reduces to the problem of
Riemannian space forms of positive curvature.
Renowned work of
John Nash in the 1950s considered the problem of
isometric embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
s. His work showed that such embeddings are very flexible and deformable. In his PhD thesis, Calabi had previously considered the special case of holomorphic isometric embeddings into
complex-geometric space forms. A striking result of his shows that such embeddings are completely determined by the intrinsic geometry and the curvature of the space form in question. Moreover, he was able to study the problem of existence via his introduction of the ''diastatic function'', which is a locally defined function built from
Kähler potentials and which mimics the Riemannian distance function. Calabi proved that a holomorphic isometric embedding must preserve the diastatic function. As a consequence, he was able to obtain a criterion for local existence of holomorphic isometric embeddings.
Later, Calabi studied the two-dimensional
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s (of high codimension) in round spheres. He proved that the area of topologically spherical minimal surfaces can only take on a discrete set of values, and that the surfaces themselves are classified by
rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s in a certain
hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
.
Major publications
Calabi is the author of fewer than fifty research articles. A large proportion of them have become a major part of the research literature.
Calabi's collected works were published in 2021:
*
Further reading
*
*
References
{{DEFAULTSORT:Calabi, Eugenio
1923 births
Living people
Scientists from Milan
20th-century American mathematicians
21st-century American mathematicians
Italian emigrants to the United States
Differential geometers
Putnam Fellows
Fellows of the American Mathematical Society
Institute for Advanced Study visiting scholars
Members of the United States National Academy of Sciences
Princeton University alumni
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
University of Minnesota faculty