Aubin–Yau theorem

Aubin–Yau theorem
Name	Aubin–Yau theorem
Field	Differential geometry, Complex geometry, Partial differential equations
Mathematicians	Thierry Aubin; Shing-Tung Yau
Year	1976; 1978
Related	Calabi conjecture; Kähler–Einstein metric; Monge–Ampère equation

Aubin–Yau theorem The Aubin–Yau theorem establishes existence results for Kähler–Einstein metrics and solutions of complex Monge–Ampère equations on compact Kähler manifolds, resolving cases of the Calabi conjecture through analytic methods. The theorem links techniques from elliptic partial differential equations, potential theory, and complex differential geometry, influencing later work by figures associated with the Poincaré conjecture, Hodge theory, and string theory.

Statement of the theorem

The theorem asserts that on a compact Kähler manifold with first Chern class of definite sign, there exists a Kähler–Einstein metric in the given Kähler class, provided certain obstructions vanish; it gives existence for negative and zero first Chern class and, with additional hypotheses, for positive first Chern class. In the negative case and the Ricci-flat case the results produce unique metrics solving a nonlinear complex Monge–Ampère equation, connecting to prior conjectures by Eugenio Calabi and subsequent applications in complex algebraic geometry, moduli theory, and mathematical physics.

Historical background and motivation

Motivated by Calabi's conjecture, the problem attracted attention from mathematicians working on Riemannian geometry and complex manifolds, including names associated with the Hilbert problems, the Fields Medal corpus, and developments in Hodge theory. Earlier work by Élie Cartan and Kunihiko Kodaira on complex manifolds and by Jean-Pierre Serre on sheaf cohomology framed the cohomological obstructions; analytic precedents included elliptic regularity results tied to Lars Hörmander and Louis Nirenberg. The breakthrough contributions occurred amid research streams connected to the International Congresses and institutions linked to Princeton University, Harvard University, and the University of Paris.

Outline of the proofs (Aubin and Yau)

Aubin's approach adapted continuity method ideas used in nonlinear analysis by Serge Bernstein and Ennio De Giorgi, employing a priori estimates and compactness methods reminiscent of John Nash's and Richard Hamilton's techniques. Aubin obtained existence in the positive first Chern class case under assumptions related to Matsushima's obstruction and Lie algebraic criteria associated with Élie Cartan's theory. Yau's proof solved the general complex Monge–Ampère equation using continuity method, maximum principle, and delicate second and third order estimates tied to work of Charles Fefferman and Lars Hörmander; Yau's argument provided uniqueness and regularity statements resonant with results by Henri Poincaré and Andrey Kolmogorov in PDE theory.

Key analytical tools and techniques

The proofs rely on the continuity method, a priori C^0, C^2, and higher order estimates, the maximum principle, and elliptic regularity theory developed by Agmon, Douglis, and Nirenberg. The complex Monge–Ampère equation invokes pluripotential techniques related to Kiyoshi Oka and Giuseppe Tomassini, while Sobolev and Moser iteration estimates trace to Jürgen Moser and Louis Nirenberg. Additional analytic constructs include the Schauder estimates associated with George Schauder, the Calabi estimate framework conceived by Eugenio Calabi, and geometric invariant theory considerations from David Mumford and Michael Atiyah that address stability conditions in the positive first Chern class case.

Applications and consequences

The theorem resolved central cases of the Calabi conjecture and produced Ricci-flat Kähler metrics on Calabi–Yau manifolds, impacting work by Pierre Deligne, Phillip Griffiths, and Nicholas Katz on Hodge structures and mirror symmetry programs advanced by Cumrun Vafa and Edward Witten in string theory. It underpins existence results used in Yau's proof of the Calabi–Yau theorem applied to Donaldson–Thomas theory and influences developments in birational geometry explored by Shigefumi Mori and Jean-Pierre Serre. Consequences extend to moduli problems studied by Alexander Grothendieck, geometric quantization themes with Simon Donaldson, and compactification scenarios in mathematical physics considered by Edward Frenkel and Michael Green.

Extensions and generalizations

Subsequent generalizations include results for singular Kähler varieties treated by János Kollár and Claire Voisin, twisted and conical Kähler–Einstein metrics pursued by Gábor Székelyhidi and Song Sun, and parabolic flows like the Kähler–Ricci flow developed by Richard Hamilton and Grigori Perelman with relations to the Poincaré conjecture. Work on stability conditions linking to the Yau–Tian–Donaldson conjecture involves Gang Tian, Simon Donaldson, and S.-T. Yau, while noncompact and complete settings connect to Rafe Mazzeo and Jeff Streets. Further research explores analytic techniques from Terence Tao and Luis Caffarelli applied to degenerate Monge–Ampère equations and pluripotential advances by Vincent Guedj and Ahmed Zeriahi.

Category:Complex differential geometry