Thierry Aubin (1942–2009): The Architect of Geometric Analysis
Thierry Aubin was a towering figure in 20th-century mathematics, specifically within the realm of differential geometry and partial differential equations (PDEs). He is best remembered for his pioneering work on the Yamabe Problem and for being a founding father of "Nonlinear Analysis on Manifolds." His work bridged the gap between the abstract world of curved spaces (geometry) and the rigorous world of calculus-based equations (analysis), providing the tools necessary to understand the shape of the universe and the behavior of physical fields.
1. Biography: From Béziers to the Académie des Sciences
Thierry Aubin was born on May 6, 1942, in Béziers, France. A product of the elite French "Grandes Écoles" system, he entered the École Polytechnique in 1961. This institution, known for its rigorous mathematical and scientific curriculum, provided the foundation for his analytical prowess.
After graduating, Aubin pursued doctoral research under the supervision of the legendary André Lichnerowicz, one of the most influential mathematical physicists of the era. Aubin defended his thesis in 1969, focusing on the relationship between curvature and the Laplacian operator on Riemannian manifolds.
He spent the vast majority of his career as a Professor at the Université Pierre et Marie Curie (Paris VI), now part of Sorbonne University. His career was marked by a steady ascent into the upper echelons of the French scientific establishment, culminating in his election to the French Academy of Sciences (Académie des Sciences) in 2003. Aubin remained active in research and teaching until his death on March 16, 2009.
2. Major Contributions: Solving the Unsolvable
Aubin’s work focused on how to find "best" or "ideal" metrics (ways of measuring distance) on a given geometric shape.
The Yamabe Problem
Aubin’s most famous contribution is his work on the Yamabe Problem. In 1960, Hidehiko Yamabe claimed to have proven that every smooth, compact Riemannian manifold can be "reshaped" (via a conformal transformation) to have a constant scalar curvature. Essentially, Yamabe argued that any lumpy, irregular shape could be smoothed out into a perfectly uniform one.
However, in 1968, Neil Trudinger discovered a fatal flaw in Yamabe’s proof. The problem became one of the most famous challenges in geometry. In 1976, Aubin provided the breakthrough. He proved that the conjecture was true for all manifolds of dimension $n \ge 6$ that are not "locally conformally flat." He introduced the "Aubin Inequality," a sophisticated use of Sobolev spaces, to show that the energy required to smooth the manifold was lower than a specific threshold. His work covered the vast majority of cases, leaving only a few specific scenarios that were later completed by Richard Schoen in 1984.
Nonlinear Analysis on Manifolds
Aubin was a pioneer in applying nonlinear analysis to curved spaces. Before him, many analysts worked on "flat" Euclidean space ($R^n$). Aubin showed how to adapt these complex tools—specifically Sobolev inequalities—to Riemannian manifolds. This work laid the groundwork for modern geometric flows, including the Ricci Flow used by Grigori Perelman to solve the Poincaré Conjecture.
The Monge-Ampère Equations and the Calabi Conjecture
Aubin worked extensively on the complex Monge-Ampère equations. He was a key player in the race to solve the Calabi Conjecture, which concerns the existence of Kähler-Einstein metrics. While Shing-Tung Yau eventually provided the full proof (earning a Fields Medal), Aubin independently proved the conjecture for the case of negative first Chern class, a massive achievement in its own right.
3. Notable Publications
Aubin was known for writing with extreme precision. His books remain standard references for graduate students and researchers:
- "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire" (1976): The seminal paper in Journal de Mathématiques Pures et Appliquées that solved the bulk of the Yamabe problem.
- "Nonlinear Analysis on Manifolds: Monge-Ampère Equations" (1982): This book became the "bible" for researchers entering the field of geometric analysis.
- "Some Nonlinear Problems in Riemannian Geometry" (1998): A comprehensive expansion of his earlier work, detailing the interplay between PDEs and geometry.
4. Awards & Recognition
While Aubin did not receive the Fields Medal (often awarded to younger mathematicians), his contributions were recognized by the highest honors in French science:
- Prix Servant (1982): Awarded by the French Academy of Sciences for outstanding research in mathematics.
- Election to the Académie des Sciences (2003): This is the highest honor for a French scientist, recognizing a lifetime of significant contribution to human knowledge.
- Invited Speaker at the International Congress of Mathematicians (ICM): He was invited to speak at the ICM in 1978 (Helsinki), a mark of global prestige in the math community.
5. Impact & Legacy
Thierry Aubin is often cited as one of the founders of Geometric Analysis. This field is now one of the most active areas of mathematics, crucial for modern physics (General Relativity) and topology.
His legacy lives on through the Aubin Inequality and the techniques he developed for handling "critical exponents" in differential equations. Every time a mathematician uses a Sobolev embedding on a manifold, they are walking the path Aubin cleared. Furthermore, his work on the Yamabe problem demonstrated that global geometric properties could be controlled by local analytical estimates, a philosophy that dominates the field today.
6. Collaborations & Mentorship
Aubin was a central figure in the "French School" of geometry. He maintained a long-standing intellectual dialogue with colleagues at the Institut des Hautes Études Scientifiques (IHÉS) and the École Normale Supérieure.
He was also a dedicated mentor. He supervised numerous PhD students who went on to become prominent mathematicians, including Emmanuel Hebey, who continued Aubin's work on nonlinear analysis and Sobolev spaces. His teaching style was described as rigorous and demanding, reflecting his belief that mathematical truth required absolute clarity.
7. Lesser-Known Facts
- The "Almost" Fields Medal: Within the mathematical community, it is often discussed how close Aubin came to the Fields Medal for his Yamabe work. Because Richard Schoen completed the final, difficult "low-dimensional" cases later, the credit for the problem is often shared, but Aubin’s 1976 paper is universally recognized as the intellectual turning point.
- A Man of the "Midi": Despite his long career in the grey streets of Paris, Aubin maintained a deep connection to his roots in the South of France (the Midi). He was known for his directness and a certain Mediterranean warmth that contrasted with the often-stuffy atmosphere of Parisian academia.
- Rigorous Simplicity: Aubin was known for his ability to take incredibly dense analytical proofs and find the "right" inequality that made everything click. His peers often remarked that he had a "physical intuition" for how functions behaved on curved surfaces.
Thierry Aubin’s life was a testament to the power of persistence. By tackling a problem that many thought was already solved (but was actually broken), he opened a new frontier in mathematics that continues to yield discoveries today.