Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on

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 poin ...

and non-linear

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold can be conformally rescaled to produce a manifold 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 geometry ...

. Along with Yau, he also showed that

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...

s with negative first

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

es always admit Kähler–Einstein metrics, a result closely related to 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 Oswa ...

. The latter result, established by Yau, provides the largest class of known examples of compact

Einstein manifold In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition i ...

s. Aubin was the first mathematician to propose the

Cartan–Hadamard conjecture In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, k ...

. Aubin was a visiting scholar at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

in 1979. He was elected to the

Académie des sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the ...

in 2003.

Research

In 1970, Aubin established that any

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

of dimension larger than two has a

Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...

of negative

. Furthermore, he proved that a Riemannian metric of nonnegative

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 measure ...

can be deformed to positive Ricci curvature, provided that its Ricci curvature is strictly positive at one point. In the same year, Aubin introduced an approach to the

, in the field of

Kähler geometry 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 ...

, via the

calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...

. Later, in 1976, Aubin established the existence of

Kähler–Einstein metric In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The ...

s on

s whose

first Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...

is negative. Independently,

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 ...

proved the more powerful Calabi conjecture, which concerns the general problem of prescribing the Ricci curvature of a Kähler metric, via non-variational methods. As such, the existence of Kähler–Einstein metrics with negative first Chern class is often called the ''Aubin–Yau theorem''. After learning Yau's techniques from

Jerry Kazdan Jerry Lawrence Kazdan (born 31 October 1937 in Detroit, Michigan) is an American mathematician noted for his work in differential geometry and the study of partial differential equations. His contributions include the Berger–Kazdan comparis ...

, Aubin found some simplifications and modifications of his work, along with Kazdan and

Jean-Pierre Bourguignon Jean-Pierre Bourguignon (born 21 July 1947) is a French mathematician, working in the field of differential geometry. Biography Born in Lyon, he studied at École Polytechnique in Palaiseau, graduating in 1969. For his graduate studies he went t ...

. Aubin made a number of fundamental contributions to the study of

Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...

s on Riemannian manifolds. He established Riemannian formulations of many classical results for Sobolev spaces, such as the equivalence of various definitions, the density of various subclasses of functions, and the standard embedding theorems. In one of Aubin's best-known works, the analysis of the optimal constant in the

Sobolev embedding theorem In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...

was carried out. Along with similar results for the

Moser–Trudinger inequality In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provi ...

, Aubin later proved improvements of the optimal constants when the functions are assumed to satisfy certain orthogonality constraints. Such results are naturally applicable to many problems in the field of

geometric analysis Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...

. Aubin considered the

Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...

on conformal deformation to constant scalar curvature, which Yamabe had reduced to a problem in the calculus of variations. Following prior work of

Neil Trudinger Neil Sidney Trudinger (born 20 June 1942) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B.Sc at the University of New England (Australia) ...

, Aubin was able to resolve the problem in high dimensions under the condition that the Weyl curvature is nonzero at some point. The key of Aubin's analysis is essentially local, with an estimate on the geometry of the

Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential ...

based on the Weyl curvature. The more subtle case of

locally conformally flat A (pseudo-Riemannian manifold, pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to Flat manifold, flat space by a conformal transformation. In practice, the Metric tensor, metric g of the manifold ...

manifolds, along with the low-dimensional case, was later established by

Richard Schoen Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984. Career Born in Celina, Ohio, and a 1 ...

as an application of Schoen and Yau's

positive mass theorem The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...

. All of the results outlined here, along with many others, were absorbed into Aubin's book ''Some Nonlinear Problems in Riemannian Geometry'', which has become a basic part of the research literature.This book is an expansion of Aubin's prior book ''Nonlinear analysis on manifolds. Monge–Ampère equations''.

Major Publications

Articles. Aubin was the author of around sixty research papers. The following, among the best-known, are outlined above. * * * * * * * Books *
Expansion of: :: *

References

External links

*
Obituary
on the SMF ''Gazette'' {{DEFAULTSORT:Aubin, Thierry 1942 births 2009 deaths 20th-century French mathematicians Differential geometers Members of the French Academy of Sciences Institute for Advanced Study visiting scholars École Polytechnique alumni 21st-century French mathematicians