Yamabe ProblemEdit
The Yamabe problem asks whether every conformal class of Riemannian metrics on a compact manifold contains a metric of constant scalar curvature. Originating in the work of the Japanese mathematician Hide Yamabe in the 1960s, the problem sits at the heart of geometric analysis, linking the shape of a space to a single global curvature invariant. It translates a geometric question into a nonlinear elliptic partial differential equation (PDE) and was resolved only after several deep insights and technical refinements by different researchers over two decades. The resolution established a canonical way to select a representative metric in each conformal class, providing a powerful tool for comparing and contrasting different geometric structures.
In broad terms, the problem asks for a metric within the same conformal class as a given metric g on a compact manifold M such that its scalar curvature is constant. The scalar curvature encodes, in a compact way, how the manifold bends: a positive scalar curvature suggests a spherical-type geometry, zero indicates flatness, and negative values point to hyperbolic-type geometry. Finding a constant-curvature representative allows one to normalize the geometry of M and to study how curvature interacts with topology and global analysis. The problem is inherently tied to the conformal Laplacian and variational methods, and its solution rests on controlling how curvature transforms under conformal changes of the metric.
Historical background
The Yamabe problem is named for Hide Yamabe, who conjectured in the early 1960s that every compact Riemannian manifold admits a metric conformally related to the given one with constant scalar curvature. Yamabe’s initial argument proposed a variational reduction to a nonlinear equation, but it contained gaps that needed to be addressed. The key difficulty lay in the critical exponent that arises from the Sobolev embedding, which prevents a straightforward application of standard compactness arguments.
In the late 1960s and 1970s, progress came from a sequence of refinements. Neil Trudinger made a pivotal contribution by clarifying the role of the critical exponent in the variational setup and by establishing a crucial compactness result in certain settings. Thierry Aubin pushed the analysis further, demonstrating existence of solutions for many manifolds and, in particular, showing that the only remaining obstruction in some cases is the sphere with its large conformal group. The combination of these insights laid the groundwork for a complete resolution.
Richard Schoen finally completed the proof in 1984, addressing the remaining delicate case and bridging the gap left by earlier work. A central ingredient in his argument was the positive mass theorem, developed independently by Schoen and Yau and later connected to broader geometric analysis. The full resolution shows that every compact manifold indeed carries a metric in each conformal class with constant scalar curvature.
Beyond the core existence result, mathematicians have since extended the ideas to related problems, including variations of the equation, higher-order curvatures, and problems on noncompact manifolds or with boundary.
Mathematical formulation
Let M be a smooth compact manifold of dimension n ≥ 3 equipped with a Riemannian metric g. If we perform a conformal change of metric g_hat = u^{4/(n-2)} g with a positive function u on M, the scalar curvature transforms according to a conformally invariant operator, the conformal Laplacian L_g, defined by L_g = -a_n Δg + R_g, where Δ_g is the Laplace-Beltrami operator, R_g is the scalar curvature of g, and a_n = 4(n−1)/(n−2). The Yamabe equation expresses the requirement that the transformed metric g_hat has constant scalar curvature R{g_hat}: -a_n Δg u + R_g u = R{g_hat} u^{(n+2)/(n−2)}. Thus finding a metric of constant scalar curvature within the conformal class of g is tantamount to solving this nonlinear elliptic PDE for a positive function u. Equivalently, one may seek a constant λ such that L_g u = λ u^{(n+2)/(n−2)} with u > 0.
A variational viewpoint considers the Yamabe functional on the conformal class: Y_g(u) = ∫_M (a_n |∇_g u|^2 + R_g u^2) dvol_g / (∫_M |u|^{2n/(n−2)} dvol_g)^{(n−2)/n}, and the Yamabe constant of the conformal class [g] as the infimum of this functional over positive u. The problem is to show that this infimum is achieved by some positive minimizer, giving a metric with constant scalar curvature in the class.
Invariants and variational approach
The work on the Yamabe problem blends geometry, analysis, and variational methods. The key object is the Yamabe invariant (or constant) of a manifold, which captures the best possible scalar curvature in a given conformal class after the scaling normalization. The critical exponent in the Sobolev embedding is what makes the analysis delicate: the lack of compactness at the critical exponent leads to phenomena such as bubbling, where maximizing sequences concentrate energy at points and resemble scaled copies of the standard sphere’s metric.
Several technical tools are central in the proof: - The conformal Laplacian L_g and its spectral properties. - The concentration-compactness principle, introduced by Pierre-Louis Lions, to control loss of compactness due to the critical exponent. - Test function arguments that compare the manifold to the sphere, exploiting the known optimal constants on the standard sphere S^n. - The positive mass theorem, whose implications help rule out certain blow-up scenarios and complete the existence argument in the delicate cases.
By combining these ideas, the variational problem yields a minimizer in every conformal class, except for the sphere where the conformal group is large enough to create noncompactness; the final step shows that, even in this case, a constant-scalar-curvature representative exists.
The sphere and the positive mass theorem
The sphere S^n plays a special role. Its conformal group is large, and the standard sphere already realizes the optimal constant in the Sobolev inequality. The challenge is to show that the infimum in other manifolds is strictly less than or equal to the sphere’s value and to preclude concentration phenomena that could prevent attainment. The positive mass theorem provides a crucial geometric input: it rules out certain pathological blow-up configurations and ensures that the limiting process produces a genuine minimizer. This interplay between local PDE analysis and global geometric invariants is a hallmark of the Yamabe problem and a key reason why the proof required tools from global differential geometry.
Generalizations and related problems
Since the resolution of the Yamabe problem, several related directions have been explored: - Extensions to manifolds with boundary, where the problem involves prescribed boundary curvature and conformal changes that affect both interior and boundary data. - Higher-order and fractional variants, such as the prescribed Q-curvature problem and fractional Yamabe problems, which replace the conformal Laplacian with more general operators reflecting different geometric quantities. - The Yamabe flow, a geometric evolution equation designed to deform an initial metric toward one with constant scalar curvature within its conformal class, providing a dynamic approach to the static problem. - Connections to CR geometry and the CR Yamabe problem, which studies analogous questions on boundaries of complex manifolds.
Controversies and debates
The development of the Yamabe problem is notable for its methodological debates and the rigorous standards that eventually prevailed. The initial proof by Yamabe began a line of inquiry that needed refinement and, at times, replacement with more robust arguments. The exchange among Trudinger, Aubin, and Schoen illustrates how mathematical communities converge on rigorous foundations: what began as an elegant variational idea required delicate compactness analyses, sharp inequalities, and global geometric input to become a complete theorem. The use of the positive mass theorem, in particular, highlights how results from geometric analysis and mathematical physics can illuminate pure geometric questions. The final resolution stands as a testament to the power of cross-disciplinary methods in mathematics and the value of rigorous, verification-driven progress.