Yamabe problem

Yamabe problem
Name	Yamabe problem
Field	Differential geometry, Partial differential equation, Riemannian geometry
Originated	1960s
Proposer	Hidehiko Yamabe
Solved	1980s
Keywords	Conformal geometry, scalar curvature, Sobolev inequality, conformal Laplacian

Yamabe problem The Yamabe problem asks whether a given compact Riemannian manifold can be conformally rescaled to produce a metric of constant scalar curvature. It connects analysis on Hilbert space, geometric invariants like the Yamabe invariant and analytic tools from the Sobolev inequality to study existence and compactness of solutions to a nonlinear elliptic Partial differential equation on manifolds. Resolution of the problem involved contributions from mathematicians working in Japan, France, the United States, and Russia and catalyzed developments in geometric analysis and the study of conformal invariants.

Introduction

The problem originates in the work of Hidehiko Yamabe and concerns a compact smooth manifold M with an initial Riemannian metric g and associated scalar curvature R_g. One seeks a positive smooth function u so that the metric u^{4/(n-2)} g (for dimension n ≥ 3) attains constant scalar curvature, turning the geometric question into an analytic existence problem for a critical exponent nonlinear equation related to the Yamabe functional. Influential figures in the subject include Richard Schoen, Trudinger, Aubin, Kazdan, Warner, Schoen-Yau, and Escobar, whose work connected the problem with the positive mass theorem and compactness issues studied by Brendle and Mari. Institutions central to the work include Princeton University, Institut des Hautes Études Scientifiques, University of California, Berkeley, and Kyoto University.

Historical development and resolution

Hidehiko Yamabe first proposed a variational approach in the 1960s while at Tokyo University of Science; subsequent counterexamples and corrections led to a long chain of contributions. Trudinger at University of Oxford corrected earlier gaps, and Aubin at Université Paris-Sud identified critical thresholds using sharp constants from the Sobolev inequality, introducing test functions modeled on the round sphere and relating to the Yamabe invariant. Richard Schoen at Stanford University completed the program using tools from the positive mass theorem by Richard Schoen and Shing-Tung Yau and techniques connected to work at Courant Institute and Harvard University. Further refinements and compactness results were obtained by Brendle at Princeton University and Khuri with collaborators, while boundary versions were developed by Escobar at University of California, Riverside. The resolution influenced later work by Perelman in Ricci flow contexts and intersected with developments by Kazdan and Warner on prescribing curvature problems.

Mathematical formulation

Given a compact n-dimensional manifold M (n ≥ 3) with metric g and scalar curvature R_g, seek a positive function u ∈ C^{∞}(M) solving the conformal scalar curvature equation L_g u = λ u^{(n+2)/(n-2)}, where L_g is the conformal Laplacian (a combination of the Laplace-Beltrami operator and R_g) and λ is the target constant scalar curvature. The variational formulation minimizes the normalized total scalar curvature functional, often written via the Yamabe functional involving integrals in the Sobolev space H^1(M). Existence and uniqueness hinge on comparison with the model case of the round sphere S^n and the sharp Sobolev constant from Aubin's inequality; when the Yamabe invariant is below that of S^n one obtains existence by direct minimization, while equality or concentration cases require delicate blow-up analysis and use of the positive mass theorem.

Methods and techniques

Key analytic techniques include concentration-compactness, blow-up analysis, test function construction inspired by the Green's function of L_g, and sharp Sobolev inequalities introduced by Thierry Aubin. Geometric tools involve conformal deformation, the study of conformal Killing fields, and the application of the positive mass theorem proven using minimal surface techniques by Schoen and Yau. PDE methods draw from elliptic regularity at Courant Institute, maximum principle ideas going back to Charles B. Morrey and variational methods developed in the tradition of John Nash and Ennio De Giorgi. Boundary problems incorporate the conformal boundary and require adaptations using the Dirichlet problem and conformal invariants studied by Escobar and Li Ma.

Examples and special cases

- The round sphere S^n: explicit solution families arise from conformal automorphisms of S^n studied by Élie Cartan and Henri Poincaré; the sphere realizes the maximal Yamabe invariant. - Manifolds with negative Yamabe invariant: constant negative scalar curvature metrics exist by direct minimization; examples include compact hyperbolic manifolds constructed via the Mostow rigidity context and arithmetic lattices in SO(n,1). - Locally conformally flat manifolds: studied by Schoen and Mazzeo, where classification simplifies via the developing map and holonomy groups from Kleinian groups theory. - Manifolds with boundary: Escobar-type problems require prescribing constant mean curvature on the boundary, with notable work at Rutgers University and University of Chicago addressing existence and compactness.

Related problems and generalizations

Generalizations include the prescribed scalar curvature problem studied by Kazdan and Warner, the sigma_k-Yamabe problem connected to fully nonlinear curvature operators and explored by Guan and Viaclovsky, and the CR Yamabe problem in pseudohermitian geometry developed by Jerison and Lee. Other directions relate to the study of conformal invariants like the Q-curvature (pursued by Branson and Chang), the Paneitz operator, and variational problems in conformal geometry influenced by work at ETH Zurich and Columbia University. Connections with geometric flows include applications of Ricci flow techniques used by Perelman and analyses of compactness in moduli spaces considered by researchers at Stanford University and Massachusetts Institute of Technology.

Category:Differential geometry