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39-579: 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,

78-419: 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

117-410: A constant. Proving that the set of possible F is open (in the set of smooth functions with average value 1) involves showing that if it is possible to solve the equation for some F , then it is possible to solve it for all sufficiently close F . Calabi proved this by using the implicit function theorem for Banach spaces : in order to apply this, the main step is to show that the linearization of

156-486: 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 . More broadly, Calabi introduced

195-435: A solution φ. In order to do this, Yau finds some a priori bounds for the functions φ i and their higher derivatives in terms of the higher derivatives of log( f i ). Finding these bounds requires a long sequence of hard estimates, each improving slightly on the previous estimate. The bounds Yau gets are enough to show that the functions φ i all lie in a compact subset of a suitable Banach space of functions, so it

234-604: A special case of Calabi's conjecture, since the 'right-hand side' of the Kähler–Einstein problem depends on the 'unknown' metric, thereby placing the Kähler–Einstein problem outside the domain of prescribing Ricci curvature. However, Yau's analysis of the complex Monge–Ampère equation in resolving the Calabi conjecture was sufficiently general so as to also resolve the existence of Kähler–Einstein metrics of negative scalar curvature. The third and final case of positive scalar curvature

273-474: 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

312-477: A university or college in Pennsylvania is a stub . You can help Misplaced Pages by expanding it . Calabi conjecture More precisely, Calabi's conjecture asserts the resolution of the prescribed Ricci curvature problem within the setting of Kähler metrics on closed complex manifolds. According to Chern–Weil theory , the Ricci form of any such metric is a closed differential 2-form which represents

351-596: Is a quadratic polynomial. By interpreting the problem as one of affine geometry , 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 , 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

390-421: Is a solution. The idea of the continuity method is to show that it can be solved for all f {\displaystyle f} by showing that the set of f {\displaystyle f} for which it can be solved is both open and closed. Since the set of f {\displaystyle f} for which it can be solved is non-empty, and the set of all f {\displaystyle f}

429-501: Is connected, this shows that it can be solved for all f {\displaystyle f} . The map from smooth functions to smooth functions taking φ {\displaystyle \varphi } to F {\displaystyle F} defined by is neither injective nor surjective. It is not injective because adding a constant to φ {\displaystyle \varphi } does not change F {\displaystyle F} , and it

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468-463: Is indeed an isomorphism. This is done in several steps, described below. Proving that the solution is unique involves showing that if then φ 1 and φ 2 differ by a constant (so must be the same if they are both normalized to have average value 0). Calabi proved this by showing that the average value of is given by an expression that is at most 0. As it is clearly at least 0, it must be 0, so which in turn forces φ 1 and φ 2 to differ by

507-457: Is not surjective because F {\displaystyle F} must be positive and have average value 1. So we consider the map restricted to functions φ {\displaystyle \varphi } that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive F = e f {\displaystyle F=e^{f}} with average value 1. Calabi and Yau proved that it

546-456: Is therefore equivalent to the following problem: This is an equation of complex Monge–Ampère type for a single function φ {\displaystyle \varphi } . It is a particularly hard partial differential equation to solve, as it is non-linear in the terms of highest order. It is easy to solve it when f = 0 {\displaystyle f=0} , as φ = 0 {\displaystyle \varphi =0}

585-403: The ∂ ∂ ¯ {\displaystyle \partial {\bar {\partial }}} -lemma , any other Kähler form in the same de Rham cohomology class is of the form for some smooth function φ {\displaystyle \varphi } on M {\displaystyle M} , unique up to addition of a constant. The Calabi conjecture

624-462: 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

663-608: 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 solutions introduced by Michael Crandall and Pierre-Louis Lions . By extending

702-685: 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. Calabi and Beno Eckmann discovered the Calabi–Eckmann manifold in 1953. It is notable as a simply-connected complex manifold which does not admit any Kähler metrics . Inspired by recent work of Kunihiko Kodaira , Calabi and Edoardo Vesentini considered

741-550: The Order of Merit of the Italian Republic . Calabi married Giuliana Segre in 1952, with whom he had a son and a daughter. He died on September 25, 2023, aged 100. 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 ,

780-520: The University of Minnesota in 1955, where he became a full professor in 1960. 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 1994, Calabi assumed emeritus status, and in 2014 the university awarded him an honorary doctorate of science . In 1982, Calabi

819-474: The continuity method . This involves first solving an easier equation, and then showing that a solution to the easy equation can be continuously deformed to a solution of the hard equation. The hardest part of Yau's solution is proving certain a priori estimates for the derivatives of solutions. Suppose that M {\displaystyle M} is a complex compact manifold with a Kähler form ω {\displaystyle \omega } . By

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858-465: The first Chern class . Calabi conjectured that for any such differential form R , there is exactly one Kähler metric in each Kähler class whose Ricci form is R . (Some compact complex manifolds admit no Kähler classes, in which case the conjecture is vacuous.) In the special case that the first Chern class vanishes, this implies that each Kähler class contains exactly one Ricci-flat metric . These are often called Calabi–Yau manifolds . However,

897-459: The strong maximum principle of Eberhard Hopf to his notion of viscosity solutions, Calabi was able to use his Laplacian comparison theorem to extend recent results of Joseph Keller and Robert Osserman to Riemannian contexts. Further extensions, based on different uses of the maximum principle , were later found by Shiu-Yuen Cheng and Yau, among others. In parallel to the classical Bernstein problem for minimal surfaces , Calabi considered

936-458: 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

975-689: The analogous problem for maximal surfaces , 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 which are defined on all of Euclidean space and with 'right-hand side' equal to one. Konrad Jörgens had earlier studied this problem for functions of two variables, proving that any solution

1014-499: 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

1053-410: The differential operator above is invertible. This is the hardest part of the proof, and was the part done by Yau. Suppose that F is in the closure of the image of possible functions φ. This means that there is a sequence of functions φ 1 , φ 2 , ... such that the corresponding functions F 1 , F 2 ,... converge to F , and the problem is to show that some subsequence of the φs converges to

1092-608: 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

1131-605: 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 an influence on the later work of George Mostow and Grigori Margulis , who established their global rigidity results out of attempts to understand infinitesimal rigidity results such as Calabi and Vesentini's, along with related works by Atle Selberg and André Weil . Calabi and Lawrence Markus considered

1170-407: 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

1209-448: 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

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1248-442: The problem of Riemannian space forms of positive curvature. Work of John Nash in the 1950s considered the problem of isometric embeddings . 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

1287-465: The problem of space forms of positive curvature in Lorentzian geometry . Their results, which Joseph A. Wolf considered "very surprising", assert that the fundamental group must be finite, and that the corresponding group of isometries of de Sitter spacetime (under an orientability condition) will act faithfully by isometries on an equatorial sphere. As such, their space form problem reduces to

1326-473: The term is often used in slightly different ways by various authors — for example, some uses may refer to the complex manifold while others might refer to a complex manifold together with a particular Ricci-flat Kähler metric. This special case can equivalently be regarded as the complete existence and uniqueness theory for Kähler–Einstein metrics of zero scalar curvature on compact complex manifolds. The case of nonzero scalar curvature does not follow as

1365-421: Was able to obtain a criterion for local existence of holomorphic isometric embeddings. Later, Calabi studied the two-dimensional minimal surfaces (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 curves in a certain hermitian symmetric space . Calabi

1404-694: Was elected to the National Academy of Sciences . 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 2012, he became a fellow of the American Mathematical Society. In 2021, he was awarded Commander of

1443-415: 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 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

1482-403: Was resolved in the 2010s, in part by making use of the Calabi conjecture. Calabi transformed the Calabi conjecture into a non-linear partial differential equation of complex Monge–Ampère type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric. Yau proved the Calabi conjecture by constructing a solution of this equation using

1521-399: Was the author of fewer than fifty research articles. Calabi's collected works were published in 2021: Thomas A. Scott Professorship of Mathematics The Thomas A. Scott Professorship of Mathematics is an academic grant made to the University of Pennsylvania . It was established in 1881 by the railroad executive and financier Thomas Alexander Scott . This article about

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