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| Fefferman–Graham expansion | |
|---|---|
| Name | Fefferman–Graham expansion |
| Introduced | 1985 |
| Field | Differential geometry |
| Notable contributors | Charles Fefferman, Robin Graham |
Fefferman–Graham expansion is a formal asymptotic series for a Poincaré metric on a conformally compact manifold developed by Charles Fefferman and Robin Graham. The expansion connects notions from Charles Fefferman's analysis, Robin Graham (mathematician)'s geometric analysis, and influences across Michael Atiyah, Isadore Singer, Edward Witten, Juan Maldacena, and James Simons. It provides a bridge between classic results in Élie Cartan-style conformal geometry, constructions related to the Yau conjectures, and modern developments in Conformal Field Theory.
The Fefferman–Graham expansion arose in the context of studying complete metrics with prescribed conformal infinity on manifolds related to problems examined by Élie Cartan, Shiing-Shen Chern, S.-S. Chern, and Shing-Tung Yau. It formalizes how a Riemannian metric on the interior of a manifold with boundary is encoded by a conformal class on the boundary, concepts also central to works by Henri Poincaré, Hermann Weyl, André Weil, and later investigators like John Milnor and Michael Gromov. The expansion has become influential in research programs associated with Edward Witten's investigations, Juan Maldacena's holographic proposals, and mathematical frameworks used by Simon Donaldson and Richard Hamilton.
Given a compact manifold-with-boundary whose boundary carries a conformal class represented in the style of Élie Cartan and André Lichnerowicz, the Fefferman–Graham construction seeks a defining function and a metric satisfying the Einstein condition with negative scalar curvature analogous to the metrics studied by Shing-Tung Yau and Michael Gromov. The setup uses tools from the analysis of pseudo-differential operators developed by Lars Hörmander, Louis Nirenberg, and Elias Stein; it frames the metric as a formal power series in a boundary defining coordinate, reflecting techniques employed by Joseph J. Kohn and Louis Boutet de Monvel. The coefficients in the series are tensors on the boundary related to conformal invariants that echo constructions from Thomas Willmore and A. P. Calderón.
Existence and uniqueness of the formal series rely on linear and nonlinear elliptic theory influenced by results of Sergiu Klainerman, Dennis Sullivan, Gilles Lebeau, and the elliptic regularity frameworks of Agmon, Douglis, and Nirenberg. In even boundary dimensions obstruction tensors first identified in the expansion are parallel in spirit to obstructions studied by Michael Atiyah and Isadore Singer in index theory and by Kazuo Ueno in deformation theory. For odd boundary dimensions formal uniqueness holds to all orders, while in even boundary dimensions an explicit obstruction analogous to invariants investigated by Alberto Calderón and I. M. Singer appears at finite order.
The coefficients and obstruction tensors yield asymptotic invariants connected to scalar quantities studied in the works of James Simons, Nathan Seiberg, Alexander Polyakov, and Edward Witten. These invariants reproduce anomaly formulas appearing in Conformal Field Theory literature associated with Juan Maldacena's correspondence and calculations by Steven Weinberg and Gerard 't Hooft. Invariants obtained from the expansion correspond to conserved quantities and anomalies that parallel constructions by Noether and techniques used in the analysis by Leonard Susskind and Andrei Linde in theoretical physics.
Explicit examples include asymptotically hyperbolic metrics on the ball and perturbations thereof, building on model geometries familiar from Henri Poincaré's half-space, the Poincaré disk model, and developments by H. Poincaré and Felix Klein. Computations for low-dimensional examples connect to curvature tensors studied by Georges Valiron and classical formulae considered by Élie Cartan; explicit coefficients often reference canonical tensors appearing in the work of Thomas Willmore and the conformal invariants cataloged by Branson and Tom Branson-related literature.
Applications span conformal invariant operators introduced by Tom Branson and heat kernel techniques from Minakshisundaram and Atle Selberg. The expansion informs studies of the conformal Laplacian and Paneitz operators investigated by Stanley Paneitz and has been used in the geometric analysis programs developed by Richard Schoen and Karen Uhlenbeck. In physics the expansion underlies computations in the AdS/CFT correspondence central to Juan Maldacena's work, influencing practitioners such as Edward Witten, Steven Gubser, S. S. Gubser, and Klebanov in the study of holography, and it appears in anomaly calculations by Alexander Polyakov and Gerard 't Hooft.
Generalizations include higher-regularity and non-Einstein setups explored by Michael Eastwood, André Neves, and Robin Graham (mathematician), adaptations to singular spaces building on methods of Jesse T. Kalvak and studies of metric measure spaces related to Grigori Perelman's Ricci flow program. Recent work connects the formalism to index-theoretic contexts considered by Atiyah and analytic techniques advanced by Lars Hörmander and Terence Tao, and to broader geometric-analytic invariants investigated by Peter Sarnak and Jean-Pierre Serre.