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Yamabe problem

The Yamabe problem is a fundamental question in differential geometry that seeks to determine whether, for every compact Riemannian manifold (M, g) of dimension n \geq 3, there exists a positive smooth function u: M \to \mathbb{R} such that the conformal metric \tilde{g} = u^{4/(n-2)} g has constant scalar curvature.^[1] This problem, which generalizes the uniformization theorem for surfaces to higher dimensions, reduces to solving the nonlinear elliptic partial differential equation known as the Yamabe equation: $4\frac{n-1}{n-2} \Delta_g u + R_g u = \lambda u^{\frac{n+2}{n-2}}, where \Delta_g is the Laplace-Beltrami operator, R_g is the scalar curvature of g, and \lambda is a constant.^[2] Proposed by Japanese mathematician Hidehiko Yamabe in 1960, the problem aimed to deform any Riemannian structure on a compact manifold to one with constant scalar curvature using variational methods from elliptic partial differential equations.^[3] Yamabe's original proof contained a critical error involving an invalid compactness assumption in the Sobolev embedding theorem, which was later identified by Neil Trudinger in 1968.^[1] Trudinger repaired the argument for manifolds where the Yamabe invariant \lambda(M,) < 0, establishing existence in that case via subcritical approximations and a priori estimates.^[2] Further progress came in the 1970s and 1980s through independent contributions by Thierry Aubin and Richard Schoen. Aubin proved in 1982 that a solution exists when \lambda(M,) < \lambda(\mathbb{S}^n), the Yamabe invariant of the standard sphere, by demonstrating a positive lower bound on the best constant in the Sobolev inequality using local test functions.^[1] Schoen completed the resolution in 1984 for the remaining cases where \lambda(M,) = \lambda(\mathbb{S}^n) but M is not conformally equivalent to the sphere, employing global test functions derived from the Green's function and the positive mass theorem from general relativity to rule out bubbling phenomena and ensure compactness of minimizers.^[2] Together, these efforts affirmatively solved the Yamabe problem for all compact manifolds of dimension at least 3, with the constant scalar curvature equal to the sign of the Yamabe invariant.^[1] The resolution of the Yamabe problem has had profound implications for geometric analysis, influencing developments in scalar curvature prescription, the study of Einstein metrics, and flows like the Yamabe flow, which evolves metrics toward constant scalar curvature.^[4] It also connects to broader themes in Riemannian geometry, such as the index of the conformal Laplacian and the geometry of positive mass theorems.^[2] Extensions to manifolds with boundary, noncompact cases, and higher-order analogs like the k-Yamabe problem continue to be active areas of research.^[5]

Background and Formulation

Historical Context

The Yamabe problem originated in 1960 when Japanese mathematician Hidehiko Yamabe conjectured that every compact Riemannian manifold of dimension at least three admits a conformal metric with constant scalar curvature, motivated by efforts to address the Poincaré conjecture through geometric deformations.^[6] In his seminal paper, Yamabe claimed a proof using a sequence of approximating metrics in the conformal class—defined as the set of metrics related by positive scalar multiples of powers of the original metric—but the argument contained a critical error in the regularity estimates for the limiting solution.^[6] In 1968, Neil Trudinger identified the flaw in Yamabe's proof and provided a corrected version for cases where the scalar curvature is nonpositive, as well as for subcritical approximations where the nonlinearity is weaker than the critical Sobolev exponent. This partial resolution advanced the understanding of conformal deformations but left open the supercritical regime. Three years later, in 1971, Morio Obata contributed a rigidity result specific to the standard sphere, demonstrating that any metric conformal to the round metric with constant scalar curvature must arise from a conformal diffeomorphism of the sphere itself.^[7] The problem saw significant progress in 1976 through Thierry Aubin's variational approach, which established solvability for all compact manifolds where the Yamabe invariant—the infimum of the normalized Einstein-Hilbert functional over the conformal class—is strictly less than that of the standard sphere, applicable in dimensions three and higher. Aubin's methods exploited Sobolev inequalities and compactness arguments to minimize the functional, resolving the issue in most cases but leaving a gap when the invariant equals the sphere's value. This breakthrough shifted attention to the remaining exceptional scenarios. The full resolution came in 1984 with Richard Schoen's innovative application of the positive mass theorem from general relativity, proving that the Yamabe invariant is strictly less than the sphere's for all non-spherical compact manifolds, thereby completing the affirmative solution to Yamabe's conjecture across all dimensions.^[8] This culmination marked a major milestone in conformal geometry, influencing subsequent developments in scalar curvature prescriptions.

Problem Statement

The Yamabe problem concerns Riemannian manifolds. A Riemannian manifold is a smooth manifold M equipped with a Riemannian metric g, which is a smooth, positive-definite inner product on each tangent space T_p M varying smoothly with p \in M. The conformal class $$ of g consists of all metrics \tilde{g} = \phi g where \phi: M \to (0, \infty) is a smooth positive function.^[3] The scalar curvature \mathrm{Scal}_g of (M, g) is a pointwise function derived from the Ricci curvature tensor, providing a measure of the intrinsic curvature of the manifold. The problem seeks a metric h \in such that \mathrm{Scal}_h is constant, equal to the Yamabe constant \lambda = Y(), the infimum of the total scalar curvature normalized by volume in the conformal class.^[3] To formulate this, consider the Yamabe functional on the space of metrics in $$, defined for \dim M = n \geq 3 as

Y(g) = \frac{\int_M \mathrm{Scal}_g \, d\mathrm{vol}_g}{\left( \int_M d\mathrm{vol}_g \right)^{(n-2)/n}},

where d\mathrm{vol}_g is the volume form induced by g. The Yamabe constant is \lambda = \inf_{h \in } Y(h), and the problem is to find a minimizer h achieving constant \mathrm{Scal}_h = \lambda \left( \mathrm{vol}_h \right)^{-2/n}. Under a conformal change h = u^{4/(n-2)} g with u > 0 smooth, the scalar curvature transforms according to

\mathrm{Scal}_h = u^{-\frac{n+2}{n-2}} \left[ \frac{4(n-1)}{n-2} \Delta_g u + \mathrm{Scal}_g u \right],

where \Delta_g is the Laplace-Beltrami operator on (M, g). Normalizing so that \int_M d\mathrm{vol}_h = 1, this yields the nonlinear elliptic partial differential equation

\frac{4(n-1)}{n-2} \Delta_g u + \mathrm{Scal}_g u = \lambda u^{\frac{n+2}{n-2}}

for u > 0. This equation arises from the critical Sobolev embedding H^1_g(M) \hookrightarrow L^{2n/(n-2)}(M), where the exponent $2n/(n-2) is the critical Sobolev exponent, leading to lack of compactness and variational challenges in higher dimensions.^[3] The conjecture, posed for compact manifolds without boundary, states that every compact Riemannian manifold (M, g) of dimension n \geq 3 admits a conformal metric h \in with constant scalar curvature.^[3]

Solutions on Compact Manifolds

Special Cases

The two-dimensional analogue of the Yamabe problem on compact surfaces is prescribing constant Gaussian curvature in a given conformal class, a task resolved affirmatively by the uniformization theorem, which guarantees the existence of such a metric with curvature sign determined by the topology: positive on the sphere, zero on the torus, and negative on higher-genus surfaces.^[5] This result, established through complex analysis and the Riemann mapping theorem, provides an explicit classification of all compact Riemann surfaces up to biholomorphic equivalence.^[9] On the n-sphere S^n for n \geq 3, the standard round metric g_0 possesses constant positive scalar curvature \mathrm{Scal}_{g_0} = n(n-1), directly solving the problem in its conformal class. Any other metric conformal to g_0 can be transformed back to g_0 via a conformal diffeomorphism, as the sphere's conformal group acts transitively; stereographic projection further maps this to the Euclidean space \mathbb{R}^n, where the constant curvature condition corresponds to the pullback of the flat metric, confirming the round metric as the unique solution up to scaling.^[10] This explicit construction highlights the sphere's role as the minimizer of the Yamabe functional among all compact manifolds.^[5] For closed Einstein manifolds (M, g) of dimension n \geq 3 with positive scalar curvature \mathrm{Scal}_g > 0, where the Ricci tensor satisfies \mathrm{Ric}_g = (\mathrm{Scal}_g / n) g, the metric g already has constant scalar curvature, trivially resolving the Yamabe problem. In this case, the constant function is the first eigenfunction of the conformal Laplacian L_g = -\Delta_g + \frac{n-2}{4(n-1)} \mathrm{Scal}_g, achieving the minimum of the Yamabe functional.^[2] The n-torus T^n admits a flat metric g inherited from the Euclidean metric on \mathbb{R}^n via the quotient by \mathbb{Z}^n, yielding constant zero scalar curvature and solving the problem in this conformal class.^[11] Similarly, real projective spaces \mathbb{RP}^n inherit constant positive scalar curvature metrics from the round metric on S^n through the antipodal quotient, as this Riemannian submersion preserves the constant curvature property locally.^[12] On product manifolds or more generally those with positive Yamabe invariant, the Kazdan-Warner obstructions—integral identities that any scalar curvature function must satisfy with respect to Killing vector fields—do not arise for constant positive scalar curvature, enabling solutions via direct variational minimization of the Yamabe functional without topological impediments.^[13] These cases, where the Yamabe invariant \lambda(M,) > 0 matches that of the sphere, informed later techniques by demonstrating the absence of barriers in symmetric or positively curved settings.^[2]

General Solution

The general solution to the Yamabe problem on compact Riemannian manifolds of dimension n \geq 3 establishes the existence of a conformal metric with constant scalar curvature in every conformal class, completing the resolution of the conjecture through analytic and geometric techniques. A pivotal advance came from Thierry Aubin's 1982 analysis, which proved existence when the Yamabe invariant Y(M, ) < Y(\mathbb{S}^n), for dimensions n \geq 3.^[2] Aubin employed subcritical approximations of the Yamabe equation, leveraging Sobolev embeddings to construct nearly minimizing sequences, and introduced concentration-compactness principles to control the lack of compactness in the functional space, ensuring convergence to a solution in these cases.^[1] The conformal Laplacian operator plays a central role in reformulating the problem as a nonlinear elliptic PDE: L_g u = -a_n \Delta_g u + \Scal_g u, where a_n = \frac{4(n-1)}{n-2} is the conformal coefficient and \Scal_g is the scalar curvature. The Yamabe equation then seeks a positive solution u such that L_g u = \lambda u^{\frac{n+2}{n-2}}, where \lambda is a constant determining the sign of the scalar curvature of the conformal metric g_u = u^{\frac{4}{n-2}} g. Eigenvalue problems for L_g further inform the sign of Y(M, ), with the first eigenvalue influencing the existence strategy. Richard Schoen completed the proof in 1984 for the remaining cases where Y(M, ) = Y(\mathbb{S}^n) but M is not conformally equivalent to the standard sphere.^[1] Schoen, assuming no minimizer exists, analyzed minimizing sequences using global test functions derived from the Green's function. In potential blow-up limits, the positive mass theorem applied to resulting asymptotically flat manifolds yields a contradiction unless M is conformally the sphere, ensuring compactness of minimizers.^[2] The positive mass theorem, established by Schoen and Yau, asserts that for an asymptotically flat manifold of dimension n \leq 7 with nonnegative scalar curvature, the ADM mass is nonnegative, with equality if and only if the manifold is Euclidean space.^[14] This integral geometry result, proved using minimal surface techniques and the second variation of area, underpins Schoen's contradiction by showing that the assumed nonexistence leads to a manifold with negative mass.^[14] The resulting constant scalar curvature metric in the conformal class is unique up to positive scaling by a constant factor.

Solutions on Non-Compact Manifolds

Formulation

The Yamabe problem on non-compact manifolds is posed for a complete, connected Riemannian manifold (M, g) of dimension n \geq 3, typically assuming bounded geometry: the injectivity radius is bounded below by a positive constant, and the sectional curvature tensor along with all its derivatives are bounded above and below.^[15] Such manifolds often feature ends that are asymptotically flat or asymptotically hyperbolic, ensuring controlled behavior at infinity. The objective is to find a complete conformal metric \tilde{g} = u^{4/(n-2)} g, where u > 0 is a smooth function decaying appropriately at infinity, such that the scalar curvature \text{Scal}_{\tilde{g}} is constant, possibly zero (for asymptotically flat ends) or negative (for hyperbolic ends). Unlike the compact case, the existence of such a metric is not guaranteed on all non-compact manifolds; counterexamples were provided by Jin (1988).^[16] Solutions are known to exist under specific geometric conditions, such as bounded geometry and appropriate asymptotic behavior at infinity.^[17] This conformal change aims to achieve \text{Scal}_{\tilde{g}} = \lambda, with \lambda determined by the geometry, and u satisfying conditions like u \to 1 as the distance to a compact exhaustion goes to infinity in asymptotically flat cases. The problem reduces to solving the Yamabe equation, the same nonlinear elliptic PDE as in the compact setting:

\frac{4(n-1)}{n-2} \Delta_g u + \text{Scal}_g u = \lambda u^{\frac{n+2}{n-2}},

but with boundary conditions at infinity, such as u \geq 1 - C e^{-\alpha r} and \liminf_{r \to \infty} u \geq 1 for asymptotically hyperbolic ends with decay rate \alpha > 0, where r is the geodesic distance function.^[17] For non-compact manifolds, the Yamabe invariant Y(g) of the conformal class is defined as the infimum of the functional

Y(g) = \inf \frac{\int_M \left( \frac{4(n-1)}{n-2} |\nabla u|^2 + \text{Scal}_g u^2 \right) d\text{vol}_g}{\left( \int_M u^{2n/(n-2)} d\text{vol}_g \right)^{(n-2)/n}}

over positive smooth functions u with compact support, highlighting challenges from the lack of compactness that prevent the infimum from being attained without additional geometric assumptions.^[15] Unlike the compact case, which seeks a volume-normalized constant scalar curvature metric in the conformal class, the non-compact formulation omits volume normalization due to infinite total volume and instead prioritizes growth or decay conditions on u aligned with the ends' asymptotic structure.^[15]

Key Results

One of the early key results for the Yamabe problem on non-compact manifolds concerns the case of constant negative scalar curvature. In 1988, Aviles and McOwen established that if (M, g) is a complete Riemannian manifold of dimension n ≥ 3 with scalar curvature S_g ≤ 0 and S_g < -ε < 0 outside a compact subset M_0, then there exists a complete conformal metric \tilde{g} = u^{4/(n-2)} g with constant scalar curvature S_{\tilde{g}} = -1. This result applies in particular to \mathbb{R}^n minus a compact set under suitable decay conditions on the scalar curvature, ensuring the completeness of the new metric. The proof relies on variational methods and a removable singularity theorem to handle the behavior at infinity. For asymptotically flat (AF) manifolds, the positive mass theorem plays a central role in resolving the problem for constant zero scalar curvature. The positive mass theorem, proved by Schoen and Yau in 1979 for n=3 and extended by Witten in 1981 for higher dimensions, states that an AF manifold of dimension n ≥ 3 with nonnegative scalar curvature has nonnegative ADM mass, with equality if and only if the manifold is Euclidean space. This theorem implies that if the ADM mass is positive, the Euclidean metric cannot minimize the Yamabe functional in the conformal class, leading to the existence of a complete conformal metric with constant zero scalar curvature. Recent developments using the Yamabe flow confirm this: on an AF manifold with suitable decay (order τ > 0), the flow exists globally and converges to a complete, asymptotically flat metric of constant zero scalar curvature, preserving the positive mass.^[18] In cases where the Yamabe invariant is positive, partial resolutions employ bubbling analysis to construct solutions, particularly on AF manifolds with positive mass. For instance, manifolds with Yamabe invariant Y(g) > 0 and positive ADM mass admit complete conformal metrics of constant positive scalar curvature under additional geometric conditions, such as controlled ends; the bubbling technique analyzes concentration phenomena along minimizing sequences of the Yamabe functional to establish existence. However, general constructions for negative target scalar curvature beyond the bounded-below case often involve gluing methods or normalized Yamabe flows to handle multiple ends, yielding complete metrics with the prescribed constant.^[19] Despite these advances, the Yamabe problem remains open for complete solutions in all non-compact cases, particularly regarding the completeness of metrics when the manifold has multiple ends or when the target scalar curvature is positive without decay assumptions on the original metric.

Applications and Extensions

In Conformal Geometry

The Yamabe invariant Y(M, ) of a compact Riemannian manifold (M, g) is a key conformal invariant arising from the Yamabe problem, defined as the infimum of the Yamabe functional over all unit-volume metrics in the conformal class , or equivalently, the supremum of the constant scalar curvatures achievable by Yamabe metrics of unit volume in that class. This invariant is conformally invariant because the Yamabe functional transforms covariantly under conformal changes of metric, preserving its value across the entire class .^[20] On standard spaces, explicit computations reveal its structure: for the standard round sphere S^n with the canonical metric g_{\mathrm{std}}, Y(S^n, [g_{\mathrm{std}}]) = n(n-1) \omega_n^{2/n}, where \omega_n denotes the volume of the unit sphere; this value serves as an upper bound for Yamabe invariants on other n-manifolds, with equality if and only if the manifold is conformally diffeomorphic to the sphere. Similar computations apply to projective spaces, such as \mathbb{RP}^n, where the invariant equals that of the sphere times $2^{-2/n} due to the double covering halving the volume while preserving local geometry.^[21] The Yamabe invariant connects to broader families of conformal invariants through higher-order analogues, notably the \sigma_k-constants introduced by Viaclovsky in the context of fully nonlinear elliptic equations on conformal manifolds. These arise in the \sigma_k-Yamabe problem, which seeks a conformal metric \tilde{g} = u^{4/(n-2)} g such that the k-th elementary symmetric function \sigma_k of the eigenvalues of the trace-free Schouten tensor of \tilde{g} is constant.^[22] The associated \sigma_k-constant, defined variationally as the infimum of an integral functional over unit-volume metrics in , is conformally invariant, mirroring the Yamabe case for k=1, and extends the theory to fully nonlinear settings where the equation governs higher symmetric functions of curvature. For $1 < k \leq n/2, these constants provide obstructions to solvability and link to geometric inequalities, such as volume comparisons between manifolds and spheres.^[22] Classification results for constant scalar curvature metrics in conformal classes leverage extensions of the Obata theorem, which originally establishes that the only compact manifold with constant sectional curvature 1 is the round sphere. In the Yamabe context, these extensions imply uniqueness of the constant scalar curvature metric within certain classes: for Einstein manifolds with positive scalar curvature, any metric conformal to an Einstein metric with the same constant scalar curvature must be the original Einstein metric up to scaling, generalizing Obata's rigidity to broader conformal settings.^[23] Such uniqueness holds particularly when the Yamabe invariant is achieved, ensuring that the minimizing metric is isolated in the class, as seen in locally conformally flat manifolds or those admitting positive Yamabe metrics.^[24] Extensions of the Yamabe problem to prescribing arbitrary scalar curvatures under conformal deformations address the equation \mathrm{Scal}_{\tilde{g}} = f for a given positive function f on M, solvable via \tilde{g} = u^{4/(n-2)} g where u satisfies a semilinear elliptic PDE.^[25] This prescribed scalar curvature problem is resolved when the integral of f against a suitable test function satisfies sign conditions relative to the Yamabe invariant, with existence guaranteed for subcritical cases or when Y(M, ) < 0.^[26] More generally, allowing nonlinear prescriptions like f(\mathrm{Scal}_{\tilde{g}}) = h for a prescribed h leads to fully nonlinear equations, whose solvability draws on variational methods and a priori estimates from the linear Yamabe theory, though obstructions persist in positive Yamabe classes.^[27] In four-dimensional conformal geometry, the Yamabe problem intersects with gauge-theoretic invariants through Donaldson's work, which employs Yang-Mills theory to classify self-dual conformal structures and obstruct the existence of metrics with positive scalar curvature on certain 4-manifolds.^[28] Specifically, Donaldson's constructions of self-dual metrics in prescribed conformal classes imply that manifolds admitting such structures achieve their Yamabe invariant via positive constant scalar curvature metrics, linking topological invariants like the second Betti number to conformal rigidity.^[29] This framework extends to blow-up analyses and connected sums, where the Yamabe flow preserves conformal classes while revealing obstructions to constant curvature, influencing classifications of 4-manifolds up to diffeomorphism.^[24] Recent advances as of 2025 include a local method for resolving the Yamabe problem in dimensions at least 4 and progress on the fractional Yamabe problem via asymptotically hyperbolic Einstein manifolds.^[30]^[31]

In General Relativity

In general relativity, the Yamabe problem plays a crucial role in the conformal method for solving the Einstein constraint equations, which govern the initial data for spacetimes. The conformal method decomposes the physical metric into a conformal class and a transverse-traceless tensor, reducing the Hamiltonian constraint to the Lichnerowicz-York equation for the conformal factor \phi > 0. This semilinear elliptic equation, \Delta_g \phi - \frac{1}{8} R_g \phi + \frac{1}{8} |\hat{\sigma}|^2_g \phi^{-7} = \frac{1}{12} \tau^2 \phi^5 on a 3-manifold (M, g) with scalar curvature R_g, traceless tensor \hat{\sigma}, and mean curvature \tau, is a perturbation of the Yamabe equation for constant scalar curvature. Solutions exist under suitable conditions, such as when the background metric admits a positive Yamabe invariant, enabling the construction of initial data sets with prescribed asymptotic behavior and matter sources. The Yamabe flow, evolving the metric along its scalar curvature gradient, provides a dynamical approach to finding these conformal factors, particularly for constant mean curvature slices in asymptotically flat settings. The positive mass theorem, proved by Schoen and Yau using minimal hypersurface techniques and by Witten via spinor methods, relies on Yamabe metrics to establish nonnegative ADM mass for asymptotically flat initial data with nonnegative scalar curvature. Specifically, for a 3-manifold with asymptotically flat ends, the theorem implies that the mass is zero only if the manifold is Euclidean space, linking the existence of Yamabe metrics of constant positive scalar curvature to the rigidity of flat metrics. This has implications for black hole uniqueness: the non-existence of certain asymptotically flat solutions with positive scalar curvature horizons precludes non-Schwarzschild black holes in the stationary vacuum case, as any deviation would violate the positive mass rigidity. In higher dimensions, extensions show that horizons of stationary black holes must be of positive Yamabe type, admitting metrics of positive scalar curvature, which imposes topological restrictions like spherical connectivity in four dimensions. Bartnik's quasi-local mass definition addresses the challenge of localizing gravitational energy within finite regions of an initial data set, defined as the infimum of ADM masses over all asymptotically flat extensions outside a outer-minimizing hypersurface \Sigma. This leads to Yamabe-type problems on \Sigma, where one seeks extensions minimizing the mass while satisfying vacuum constraints outside, often involving solving a scalar curvature prescription akin to the Yamabe equation on the exterior domain. For outer-minimizing hypersurfaces enclosing apparent horizons, the quasi-local mass is nonnegative and monotonic, providing bounds that align with the positive mass theorem in the limit as \Sigma approaches infinity.^[32] Extensions to stationary spacetimes reveal obstructions from positive scalar curvature: in vacuum stationary metrics with non-degenerate horizons, the spatial slices cannot admit metrics of positive scalar curvature unless the horizon topology satisfies certain conditions, such as being diffeomorphic to a sphere in four dimensions. This follows from the dominant energy condition and the integrated form of the constraint equations, where the Yamabe invariant of the horizon cross-section must be positive to avoid contradictions with asymptotic flatness. Such obstructions underpin no-hair theorems, ensuring that stationary black holes are characterized by mass, charge, and angular momentum alone.

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