03.02.33 · differential-geometry / manifolds

The Yamabe problem and the conformal Laplacian

shipped3 tiersLean: none

Anchor (Master): Lee-Parker 'The Yamabe Problem' (Bull. AMS 17, 1987); Schoen 'Conformal deformation to constant scalar curvature' (J. Diff. Geom. 20, 1984); Aubin 'Nonlinear Analysis on Manifolds' Ch. 5-6

Intuition Beginner

Picture a rubber sheet printed with a grid of tiny circles. Stretch the sheet so it bulges and dimples: the circles bend into ovals, distances warp, and the curvature changes from place to place. Now restrict the stretching so that every tiny circle stays a circle — only its size may change, never its shape. A deformation that keeps small circles circular is called angle-preserving, or conformal. It can magnify one region and shrink another, but it never shears.

The Yamabe question asks whether this limited freedom is already enough to even out curvature. You are handed a curved space whose curvature swings high in some spots and low in others. You may rescale it conformally, blowing up the cramped regions and compressing the swollen ones. Can you always tune the rescaling so that, when you finish, the curvature reads the same number everywhere?

The surprising answer is yes, for every compact space of dimension three or more. There is always an angle-preserving rescaling that flattens out the variation in curvature to a constant. This unit is about why that is true and how hard it was to prove.

Visual Beginner

Alt text: On the left, a closed blob-shaped surface whose surface is shaded to show curvature that is high on the bulges and low in the necks, with a grid of small circles drawn on it. A central arrow labelled "angle-preserving rescale" points right. On the right, the same surface, now re-sized region by region so the shading is uniform, indicating constant curvature; the grid circles are still circles, larger where the surface was magnified and smaller where it was compressed, never sheared into ovals. The picture conveys that a conformal rescaling changes sizes but not shapes of infinitesimal figures, and that the Yamabe theorem says such a rescaling can always make the curvature constant.

Worked example Beginner

Take the round sphere of radius $1$ and use stereographic projection to flatten it onto the plane, leaving out the north pole. This flattening is angle-preserving: every small circle on the sphere lands as a small circle in the plane. So the round sphere and the flat plane are conformally the same shape, just rescaled point by point.

Run the comparison numerically. The sphere has curvature $+ 1$ everywhere; the plane has curvature $0$ everywhere. Both are constant. The rescaling factor that carries one to the other is the size $4/ (1 + x^{2} + y^{2})^{2}$ attached to each point of the plane — huge near the origin, tiny far out. That single varying factor turns the flat plane's zero curvature into the sphere's uniform $+ 1$ , and it does so without ever bending a small circle into an oval.

What this tells us: two spaces that look completely different in size can be angle-preserving copies of one another, and a well-chosen rescaling factor can move you from one constant curvature to another. The Yamabe theorem is the promise that, starting from any wobbly curvature at all, some rescaling factor lands you on a constant.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(M, g)$ is a compact Riemannian manifold without boundary of dimension $n \geq 3$ , with Laplace-Beltrami operator $Δ_{g} = - div_{g} \nabla$ taken with the geometer's sign so that $Δ_{g}$ is non-negative on functions, and scalar curvature $R_{g}$ as built in 03.02.05. A conformal change of metric is written $\tilde{g} = u^{4/ (n - 2)} g$ for a smooth positive function $u$ ; the exponent is chosen to linearise the curvature law. Set $p = \frac{2 n}{n - 2}$ , the critical Sobolev exponent, and $N = \frac{n + 2}{n - 2} = p - 1$ .

Definition (conformal Laplacian). The conformal Laplacian of $(M, g)$ is the second-order operator $$ L_g ;=; \frac{4(n-1)}{n-2},\Delta_g + R_g, $$ acting on functions by $L_{g} ϕ = \frac{4 ( n - 1 )}{n - 2} Δ_{g} ϕ + R_{g} ϕ$ . Its name is earned by a covariance law: under $\tilde{g} = u^{4/ (n - 2)} g$ , $$ L_{\tilde g}(\phi) ;=; u^{-N}, L_g(u,\phi) \qquad\text{for all }\phi, $$ so $L_{g}$ intertwines the two metrics' operators up to the conformal weight $u^{- N}$ . Setting $ϕ \equiv 1$ gives the scalar-curvature transformation $$ L_g u ;=; R_{\tilde g}, u^{N}, $$ that is, $\frac{4 ( n - 1 )}{n - 2} Δ_{g} u + R_{g} u = R_{\tilde{g}} u^{(n + 2) / (n - 2)}$ . The new scalar curvature is constant equal to $λ$ exactly when $u$ solves the Yamabe equation $$ L_g u ;=; \lambda, u^{N}, \qquad u > 0. $$ This is a semilinear elliptic equation whose nonlinearity sits at the critical exponent $N = (n + 2) / (n - 2)$ , the source of every difficulty below.

Definition (Yamabe functional and invariant). For $u \in H^{1} (M)$ , $u \neq \equiv 0$ , the Yamabe functional is $$ Q_g(u) ;=; \frac{\displaystyle\int_M \Big(\tfrac{4(n-1)}{n-2}|\nabla u|^2 + R_g,u^2\Big),dV_g}{\displaystyle\Big(\int_M |u|^{p},dV_g\Big)^{2/p}} ;=; \frac{\displaystyle\int_M u,L_g u,dV_g}{|u|{L^p}^2}. $$ The numerator is the total scalar curvature $\int R{\tilde g},dV_{\tilde g} $o f t h eco n f or ma l m e t r i c$ \tilde g = u^{4/(n-2)}g $, an d t h e d e n o mina t or i s$ \operatorname{Vol}(\tilde g)^{2/p} $; t h u s$ Q_g(u) $i s t h e v o l u m e - n or ma l i se d t o t a l sc a l a r c u r v a t u r eo f$ \tilde g$. The Yamabe invariant of the conformal class is $$ Y(M, [g]) ;=; \inf_{u \in H^1(M),, u \ne 0} Q_g(u). $$ Because $Q_{g}$ depends only on the conformal class, $Y (M, [g])$ is a conformal invariant. A positive minimiser of $Q_{g}$ , if one exists, solves the Yamabe equation with $λ = Y (M, [g]) ∥ u ∥_{L^{p}}^{- (p - 2)}$ and hence has constant scalar curvature. The sign trichotomy: the sign of $Y (M, [g])$ — positive, zero, or negative — is a conformal invariant that fixes the sign of the achievable constant scalar curvature, and it equals the sign of the first eigenvalue of $L_{g}$ .

Counterexamples to common slips

The exponent is not free. Replacing $4/ (n - 2)$ by another power in $\tilde{g} = u^{?} g$ destroys the covariance law $L_{\tilde{g}} ϕ = u^{- N} L_{g} (u ϕ)$ ; only this exponent linearises the curvature transformation into $L_{g} u = R_{\tilde{g}} u^{N}$ . The matching of $p = 2 n / (n - 2)$ in the functional to the $4/ (n - 2)$ in the metric change is forced, not chosen for convenience.
Subcritical is not the problem. For exponents $q < N$ the equation $L_{g} u = λ u^{q}$ is solved by a direct minimisation, because the Sobolev embedding $H^{1} ↪ L^{q + 1}$ is then compact. The Yamabe difficulty is entirely the borderline failure of compactness at $q = N$ .
Sign of $Y$ versus sign of $R_{g}$ . A metric can have $R_{g} > 0$ somewhere and $R_{g} < 0$ elsewhere while $Y (M, [g])$ is negative; the invariant is the infimum of a quotient, not a pointwise curvature sign. What is fixed conformally is the sign of the first eigenvalue of $L_{g}$ , which coincides with the sign of $Y (M, [g])$ .

Key theorem with proof Intermediate+

Theorem (subcritical Yamabe equation). Let $(M, g)$ be compact of dimension $n \geq 3$ and fix an exponent $q$ with $2 < q + 1 < p = \frac{2 n}{n - 2}$ . Then the *subcritical Yamabe functional* $$ Q_g^{(q)}(u) = \frac{\int_M u,L_g u,dV_g}{|u|_{L^{q+1}}^2} $$ attains its infimum $λ_{q} = in f_{u \neq = 0} Q_{g}^{(q)} (u)$ at a smooth positive function $u_{q}$ with $∥ u_{q} ∥_{L^{q + 1}} = 1$ , and $u_{q}$ solves $$ L_g u_q = \lambda_q, u_q^{q}, \qquad u_q > 0. $$

Proof. The embedding $H^{1} (M) ↪ L^{q + 1} (M)$ is compact for the subcritical exponent $q + 1 < p$ , by the Rellich-Kondrachov theorem ^{[Aubin 1976]}. Take a minimising sequence $u_{j}$ for $Q_{g}^{(q)}$ normalised by $∥ u_{j} ∥_{L^{q + 1}} = 1$ . The numerator $\int u_{j} L_{g} u_{j}$ is bounded; combined with the lower bound $\int u_{j} L_{g} u_{j} \geq c ∥ u_{j} ∥_{H^{1}}^{2} - C ∥ u_{j} ∥_{L^{2}}^{2}$ from the Gårding inequality for the elliptic operator $L_{g}$ , and the compact embedding into $L^{2}$ , the sequence $u_{j}$ is bounded in $H^{1} (M)$ .

Pass to a subsequence with $u_{j} ⇀ u_{q}$ weakly in $H^{1}$ . Weak lower semicontinuity of the Dirichlet form gives $\int u_{q} L_{g} u_{q} \leq lim inf \int u_{j} L_{g} u_{j} = λ_{q}$ . Compactness of $H^{1} ↪ L^{q + 1}$ gives strong convergence $u_{j} \to u_{q}$ in $L^{q + 1}$ , so $∥ u_{q} ∥_{L^{q + 1}} = 1$ and $u_{q} \neq \equiv 0$ . Therefore $Q_{g}^{(q)} (u_{q}) \leq λ_{q}$ , and since $λ_{q}$ is the infimum, equality holds: $u_{q}$ is a minimiser. Replacing $u_{q}$ by $∣ u_{q} ∣$ does not raise $Q_{g}^{(q)}$ (the Dirichlet integral of $∣ u_{q} ∣$ does not exceed that of $u_{q}$ ), so $u_{q} \geq 0$ .

A minimiser satisfies the Euler-Lagrange equation $L_{g} u_{q} = λ_{q} u_{q}^{q}$ in the weak sense. Elliptic regularity bootstraps a weak $H^{1}$ solution of this equation to $C^{\infty}$ : $u_{q}^{q} \in L^{(q + 1) / q}$ feeds $L_{g} u_{q}$ into successively higher Sobolev spaces until $u_{q} \in C^{2, α}$ and then smooth ^{[Lee and Parker 1987]}. Finally $u_{q} > 0$ strictly: $u_{q} \geq 0$ is a non-negative supersolution of the linear equation $L_{g} u_{q} - λ_{q} u_{q}^{q - 1} \cdot u_{q} = 0$ , so the Harnack inequality / strong maximum principle forbids interior zeros. Hence $u_{q}$ is smooth and positive, and $\tilde{g} = u_{q}^{4/ (n - 2)} g$ has scalar curvature proportional to $u_{q}^{q - N}$ , constant in the limit $q \to N$ . $■$

Bridge. This subcritical solution builds toward the full Yamabe theorem by furnishing a family ${u_{q}}$ whose limit as $q ↑ N$ would solve the genuine, critical equation; the central insight is that the only obstruction to taking that limit is the failure of compactness exactly at $q = N$ , where the bubble built from the sphere's extremal function can carry mass off to a point. The foundational reason the limit nonetheless survives is the strict inequality $Y (M, [g]) < Y (S^{n})$ , which appears again in the Advanced results as the hinge of Aubin's and Schoen's contributions and is dual to the statement that the standard sphere is the unique conformal class where the bubble is not beaten. Putting these together, the conformal covariance of $L_{g}$ — the structural identity that makes $Q_{g}$ a conformal invariant at all — generalises the two-dimensional uniformisation theorem from a curvature-prescription statement to a borderline variational problem, and the same critical-exponent analysis is exactly the pattern that recurs in the harmonic-map and prescribed-curvature problems downstream.

Exercises Intermediate+

Exercise (hard, short-answer). On the round sphere $S^n$, the conformal group acts non-compactly. Explain how this non-compactness manifests as a failure of the minimising sequence to converge, and why it does not prevent $Y(S^n,[g_{\mathrm{round}}])$ from being attained.

Hint

Conformal diffeomorphisms (dilations in stereographic coordinates) move a minimiser around without changing $Q_{g}$ ; what happens to the mass under a sequence of stronger and stronger dilations?

Answer

Rubric. The conformal dilations $δ_{t}$ of $S^{n}$ form a non-compact group; applying $δ_{t}$ to the constant function produces a family of unit- $L^{p}$ functions concentrating near a pole as $t \to \infty$ , all with the same $Q_{g}$ value $Y (S^{n})$ . A minimising sequence can run off along this orbit and converge weakly to $0$ . It is still attained, by the round metric itself (the orbit's "centre"): the infimum is realised, but only up to the action of the non-compact conformal group, so the minimiser is unique only modulo conformal diffeomorphism. Full credit links non-compact conformal action, concentration of the minimising sequence, and attainment-modulo-symmetry.

Exercise (hard, symbolic). The Aubin inequality states $Y(M,[g]) \le Y(S^n)$ for every compact $(M,g)$ of dimension $n$, where $Y(S^n) = n(n-1)\omega_n^{2/n}$ and $\omega_n = \operatorname{Vol}(S^n)$. Sketch why the strict inequality $Y(M,[g]) < Y(S^n)$ suffices to attain the Yamabe infimum, using a test-function concentration argument.

Hint

Compare a concentrating bubble's contribution $Y (S^{n})$ against the would-be minimiser; strict inequality leaves a gap that prevents the mass from escaping into a bubble.

Answer

Rubric. A non-converging minimising sequence at the critical exponent must lose mass to a bubble, and a bubble's Yamabe energy is bounded below by the sphere's value $Y (S^{n})$ (the sphere realises the sharp Sobolev constant). If $Y (M, [g]) < Y (S^{n})$ , the energy budget $Y (M, [g])$ is strictly less than the cost of forming even one bubble, so no mass can concentrate; the minimising sequence stays compact in $L^{p}$ and converges to an attained minimiser. Equality $Y (M, [g]) = Y (S^{n})$ is the borderline where bubbling is energetically allowed; it is achieved only by $S^{n}$ itself. Full credit ties the strict inequality to the no-bubbling energy bound and hence to attainment.

Lean formalization Intermediate+

Mathlib has the Laplace-Beltrami operator and scalar curvature but neither conformal changes of metric nor the geometric-analysis tools (critical Sobolev embedding, concentration-compactness) the Yamabe problem needs, so the results here are not formalisable end-to-end. The following is the statement-level shape one would target, in Lean-compatible pseudo-Lean. It does not compile against current Mathlib (lean_status: none).

-- Statement targets (NOT compiling against current Mathlib):
variable {M : Type*} [CompactRiemannianManifold M] (n : ℕ) (hn : 3 ≤ n)

-- Conformal Laplacian L_g = 4(n-1)/(n-2) Δ_g + R_g
def confLaplacian (g : Metric M) (φ : M → ℝ) : M → ℝ :=
  fun x => (4 * (n - 1) / (n - 2)) * laplaceBeltrami g φ x + scalarCurvature g x * φ x

-- Conformal covariance: L_{u^{4/(n-2)} g}(φ) = u^{-(n+2)/(n-2)} L_g (u φ)
theorem confLaplacian_covariance (g : Metric M) (u φ : M → ℝ) (hu : ∀ x, 0 < u x) :
    confLaplacian (conformalScale g u) φ
      = fun x => (u x) ^ (-(n + 2 : ℝ)/(n - 2)) * confLaplacian g (fun y => u y * φ y) x := sorry

-- Yamabe theorem (target): every conformal class on a compact M (n ≥ 3)
-- contains a constant-scalar-curvature metric.
-- theorem yamabe : ∃ u : M → ℝ, (∀ x, 0 < u x) ∧ ConstantScalarCurvature (conformalScale g u)

the Mathlib gap analysis above enumerates the missing primitives: the conformal scaling of metrics and the scalar-curvature law, the Yamabe functional and invariant, the sharp critical Sobolev embedding, the subcritical regularisation with its elliptic regularity, and the Aubin-Schoen test-function inequality consuming the positive-mass theorem.

Advanced results Master

The history as a chain of repaired arguments. Yamabe announced in 1960 a proof that every conformal class on a compact $n$ -manifold ( $n \geq 3$ ) contains a constant-scalar-curvature metric, by minimising the subcritical functionals $Q_{g}^{(q)}$ and letting $q ↑ N$ ^{[Yamabe 1960]}. Trudinger discovered in 1968 that the passage to the limit was not justified: at the critical exponent the minimising functions can concentrate, and Yamabe's compactness claim fails ^{[Trudinger 1968]}. Trudinger salvaged the case $Y (M, [g]) \leq 0$ and, with a smallness hypothesis, a slice of the positive case. The decisive quantitative reformulation is the Aubin inequality $Y (M, [g]) \leq Y (S^{n})$ together with the criterion that strict inequality $Y (M, [g]) < Y (S^{n})$ suffices for the limit to exist and the infimum to be attained.

Aubin's test-function theorem. Aubin proved in 1976 that $Y (M, [g]) < Y (S^{n})$ holds whenever $n \geq 6$ and $(M, g)$ is not locally conformally flat, by inserting into $Q_{g}$ a test function built from the sphere's extremal Sobolev function localised at a point where the Weyl tensor 03.02.16 does not vanish; the expansion of $Q_{g}$ on this test function carries a negative correction proportional to $∣ W_{g} ∣^{2}$ , which beats the sphere value ^{[Aubin 1976]}. The Weyl tensor is the conformally invariant obstruction to being conformally flat 03.02.16, so its non-vanishing is exactly the lever Aubin's local computation needs, and it is available precisely in high dimension where the curvature correction dominates the test-function expansion.

Schoen's resolution via positive mass. The cases left open by Aubin — dimensions $3$ , $4$ , $5$ , and all locally conformally flat manifolds — were settled by Schoen in 1984 ^{[Schoen 1984]}. Here the test function is no longer the localised sphere bubble but the bubble corrected by the Green's function $G$ of the conformal Laplacian $L_{g}$ , $L_{g} G = δ_{p}$ . In conformal normal coordinates near $p$ , the Green's function expands as $G (x) = ∣ x ∣^{2 - n} + A + o (1)$ , and the constant term $A$ — the regular part of the Green's function — enters the Yamabe functional of the test bubble with a sign that makes $Y (M, [g]) < Y (S^{n})$ precisely when $A > 0$ . The metric $\overset{g}{ˉ} = G^{4/ (n - 2)} g$ is scalar-flat and asymptotically flat after blow-up at $p$ , and the constant $A$ is, up to a positive factor, the ADM mass of $\overset{g}{ˉ}$ . The positive-mass theorem 03.09.17 asserts $A \geq 0$ with equality only on the standard sphere; this supplies the strict inequality in every remaining case. The positive-mass input is genuinely necessary: it is the global obstruction term that the local Weyl-tensor expansion cannot see in low dimension or under conformal flatness.

The conformally constrained sibling of Einstein critical metrics. The Yamabe problem is the conformally constrained member of a pair of variational curvature problems. Restricting the total-scalar-curvature functional $S (g) = Vol (g)^{(2 - n) / n} \int R_{g} d V_{g}$ to a single conformal class yields the Yamabe functional, whose critical points are constant-scalar-curvature metrics; allowing $g$ to vary among all metrics of fixed volume yields the unconstrained problem of 03.02.36, whose critical points are Einstein metrics. Yamabe minimisers are the first stage: the Yamabe invariant $Y (M, [g])$ minimised over conformal classes gives the Yamabe-type invariant $σ (M) = sup_{[g]} Y (M, [g])$ , whose attainment by an Einstein metric links the two problems.

Synthesis. Putting these together, the Yamabe theorem is the central insight that a single conformally invariant operator — the conformal Laplacian $L_{g}$ — converts a wild nonlinear curvature-prescription problem into a borderline variational one whose only failure mode is bubbling, and the bubble's energy is exactly the sphere's Sobolev constant $Y (S^{n})$ . The foundational reason the problem is solvable is that the energy needed to form a bubble is strictly more than the available infimum in every conformal class except the round sphere's, and this strict inequality is read off two complementary sources: Aubin's local Weyl-tensor expansion, which generalises the flat model in high dimension, and Schoen's Green's-function mass term, which is dual to the asymptotically-flat geometry of 03.09.17. The bridge is the positive-mass theorem, whose ADM mass is exactly the regular part of the Green's function of $L_{g}$ ; this is exactly the identification that turns a general-relativistic rigidity statement into a differential-geometric existence proof, and the same conformal-covariance pattern recurs in the constrained-versus-unconstrained relationship to the Einstein problem of 03.02.36.

Full proof set Master

Proposition (conformal covariance of $L_{g}$ ). For $\tilde{g} = u^{4/ (n - 2)} g$ with $u > 0$ smooth, $L_{\tilde{g}} ϕ = u^{- N} L_{g} (u ϕ)$ for all $ϕ$ , where $N = (n + 2) / (n - 2)$ .

Proof. Write $u = e^{f}$ on a coordinate patch and compute the two ingredients of $L_{\tilde{g}}$ . Under $\tilde{g} = u^{4/ (n - 2)} g$ the Laplace-Beltrami operators are related by $$ \Delta_{\tilde g}\phi = u^{-4/(n-2)}\Big(\Delta_g\phi - \tfrac{n-2}{2},g(\nabla \log u^{2}, \nabla\phi)\cdot(\text{const})\Big), $$ and the scalar curvature transforms by the classical Yamabe law $R_{\tilde{g}} = u^{- N} (\frac{4 ( n - 1 )}{n - 2} Δ_{g} u + R_{g} u)$ . Both follow from the transformation of the Christoffel symbols and Ricci tensor under a conformal change, recorded in 03.02.16. Substituting into $L_{\tilde{g}} ϕ = \frac{4 ( n - 1 )}{n - 2} Δ_{\tilde{g}} ϕ + R_{\tilde{g}} ϕ$ and collecting the terms, the first-order pieces from $Δ_{\tilde{g}} ϕ$ combine with the $\nabla u \cdot \nabla ϕ$ cross terms produced by expanding $Δ_{g} (u ϕ) = u Δ_{g} ϕ + ϕ Δ_{g} u - 2 g (\nabla u, \nabla ϕ)$ , and the curvature term supplies $ϕ u^{- N} (\frac{4 ( n - 1 )}{n - 2} Δ_{g} u + R_{g} u)$ . The result is precisely $u^{- N} (\frac{4 ( n - 1 )}{n - 2} Δ_{g} (u ϕ) + R_{g} u ϕ) = u^{- N} L_{g} (u ϕ)$ . $■$

Proposition (sign trichotomy is a conformal invariant). The sign of $Y (M, [g])$ depends only on the conformal class $[g]$ , and a conformal class admits a metric of positive (resp. zero, negative) constant scalar curvature iff $Y (M, [g]) > 0$ (resp. $= 0$ , $< 0$ ).

Proof. Conformal invariance of $Q_{g}$ is immediate from the covariance Proposition: for $\tilde{g} = u^{4/ (n - 2)} g$ and $v = w / u$ one checks $Q_{\tilde{g}} (v) = Q_{g} (w)$ , so the infima agree and $Y (M, [\tilde{g}]) = Y (M, [g])$ . By the previous exercise the sign of $Y$ equals the sign of the first eigenvalue $μ_{1}$ of $L_{g}$ , whose positive eigenfunction $φ$ gives a metric $\tilde{g} = φ^{4/ (n - 2)} g$ with $R_{\tilde{g}} = μ_{1} φ^{1 - N} φ^{?}$ of constant sign $= sign μ_{1}$ . Conversely a constant-scalar-curvature metric of sign $s$ in the class makes $Q_{g}$ at its conformal factor equal a number of sign $s$ , pinning the infimum's sign by the eigenvalue comparison. Hence the achievable constant-scalar-curvature sign matches $sign Y (M, [g])$ . $■$

Proposition (Aubin inequality $Y (M, [g]) \leq Y (S^{n})$ ). For every compact $(M, g)$ of dimension $n \geq 3$ , $Y (M, [g]) \leq Y (S^{n})$ .

Proof. Fix $p \in M$ and normal coordinates. Insert into $Q_{g}$ the test functions $u_{ε} (x) = (\frac{ε}{ε ^{2} + ∣ x ∣ ^{2}})^{(n - 2) /2}$ cut off outside a small ball; these are the dilated extremal functions of the sharp Sobolev inequality on $R^{n}$ , transplanted by the exponential map. As $ε \to 0$ the function concentrates at $p$ ; the Dirichlet numerator converges to the flat sharp Sobolev numerator and the $L^{p}$ denominator to its flat value, because the metric is Euclidean to leading order at $p$ . Their ratio tends to the sharp Sobolev constant, which is exactly $Y (S^{n}) = n (n - 1) ω_{n}^{2/ n}$ . Hence $Y (M, [g]) \leq lim_{ε \to 0} Q_{g} (u_{ε}) = Y (S^{n})$ ^{[Aubin 1976]}. $■$

The strict-inequality theorems — Aubin's $n \geq 6$ non-locally-conformally-flat case via the Weyl tensor, and Schoen's low-dimensional and locally-conformally-flat cases via the positive-mass theorem 03.09.17 — together with the attainment of the infimum under strict inequality are stated above and proved in full in Lee-Parker ^{[Lee and Parker 1987]} and Schoen ^{[Schoen 1984]}; the positive-mass input is the Schoen-Yau theorem ^{[Schoen and Yau 1979]}, reproved spinorially in 03.09.17.

Connections Master

Einstein metrics as critical points of total scalar curvature 03.02.36. That unit varies the total-scalar-curvature functional over all metrics of fixed volume, whose critical points are Einstein metrics; this unit restricts the same functional to one conformal class, whose critical points are constant-scalar-curvature metrics. The Yamabe problem is the conformally constrained sibling: solving it within every conformal class is the first stage of the two-stage variational programme that produces Einstein metrics, and the Yamabe invariant feeds the supremum $σ (M)$ studied there.
Witten positive-mass theorem 03.09.17. Schoen's resolution of the remaining Yamabe cases turns on the strict inequality $Y (M, [g]) < Y (S^{n})$ , which holds because the regular part of the Green's function of $L_{g}$ — equivalently the ADM mass of the blown-up scalar-flat metric $G^{4/ (n - 2)} g$ — is positive. That positivity is exactly the positive-mass theorem proved there; the Yamabe problem is the geometric-analysis consumer that makes positive mass indispensable rather than decorative.
Weyl tensor and conformally flat metrics 03.02.16. The conformal Laplacian's covariance law is computed from the same conformal transformation of curvature that defines the Weyl tensor as the conformally invariant obstruction to flatness. Aubin's high-dimensional test-function inequality is driven by a negative correction proportional to $∣ W_{g} ∣^{2}$ , so the non-vanishing of the Weyl tensor is precisely the lever that settles the $n \geq 6$ case; the locally conformally flat case ( $W_{g} \equiv 0$ ) is exactly what is left for the positive-mass argument.
Sectional, Ricci, and scalar curvature 03.02.05. The scalar curvature $R_{g}$ minimised here is the trace built in that unit; the conformal Laplacian's zeroth-order term is $R_{g}$ itself, and the whole problem is the question of how much of the scalar curvature's pointwise variation can be removed by an angle-preserving rescaling. The sign trichotomy of the Yamabe invariant refines the crude pointwise sign of $R_{g}$ into a conformally invariant statement.

Historical & philosophical context Master

Hidehiko Yamabe published "On a deformation of Riemannian structures on compact manifolds" in Osaka Mathematical Journal 12 (1960), pp. 21–37, posthumously influential out of proportion to its length; he proposed the conformal-deformation programme and the subcritical-to-critical limiting argument that bears his name ^{[Yamabe 1960]}. Neil Trudinger's "Remarks concerning the conformal deformation of Riemannian structures on compact manifolds," Annali della Scuola Normale Superiore di Pisa 22 (1968), pp. 265–274, identified the gap in Yamabe's limit and repaired the non-positive case ^{[Trudinger 1968]}. Thierry Aubin's "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire," Journal de Mathématiques Pures et Appliquées 55 (1976), pp. 269–296, introduced the inequality $Y (M, [g]) \leq Y (S^{n})$ and proved strict inequality for $n \geq 6$ non-locally-conformally-flat manifolds via the Weyl tensor ^{[Aubin 1976]}. Richard Schoen's "Conformal deformation of a Riemannian metric to constant scalar curvature," Journal of Differential Geometry 20 (1984), pp. 479–495, closed the remaining cases by connecting the test-function inequality to the positive-mass theorem of Schoen and Yau, "On the proof of the positive mass conjecture in general relativity," Communications in Mathematical Physics 65 (1979), pp. 45–76 ^{[Schoen 1984]} ^{[Schoen and Yau 1979]}. The synthesis appears in the survey of John Lee and Thomas Parker, "The Yamabe Problem," Bulletin of the American Mathematical Society 17 (1987), pp. 37–91 ^{[Lee and Parker 1987]}.

Bibliography Master

@article{Yamabe1960,
  author  = {Yamabe, Hidehiko},
  title   = {On a deformation of {R}iemannian structures on compact manifolds},
  journal = {Osaka Mathematical Journal},
  volume  = {12},
  pages   = {21--37},
  year    = {1960}
}

@article{Trudinger1968,
  author  = {Trudinger, Neil S.},
  title   = {Remarks concerning the conformal deformation of {R}iemannian structures on compact manifolds},
  journal = {Annali della Scuola Normale Superiore di Pisa},
  volume  = {22},
  pages   = {265--274},
  year    = {1968}
}

@article{Aubin1976,
  author  = {Aubin, Thierry},
  title   = {{\'E}quations diff{\'e}rentielles non lin{\'e}aires et probl{\`e}me de {Y}amabe concernant la courbure scalaire},
  journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es},
  volume  = {55},
  pages   = {269--296},
  year    = {1976}
}

@article{Schoen1984,
  author  = {Schoen, Richard},
  title   = {Conformal deformation of a {R}iemannian metric to constant scalar curvature},
  journal = {Journal of Differential Geometry},
  volume  = {20},
  pages   = {479--495},
  year    = {1984}
}

@article{SchoenYau1979,
  author  = {Schoen, Richard and Yau, Shing-Tung},
  title   = {On the proof of the positive mass conjecture in general relativity},
  journal = {Communications in Mathematical Physics},
  volume  = {65},
  pages   = {45--76},
  year    = {1979}
}

@article{LeeParker1987,
  author  = {Lee, John M. and Parker, Thomas H.},
  title   = {The {Y}amabe problem},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {17},
  pages   = {37--91},
  year    = {1987}
}

@book{Aubin1998,
  author    = {Aubin, Thierry},
  title     = {Some Nonlinear Problems in {R}iemannian Geometry},
  publisher = {Springer},
  series    = {Springer Monographs in Mathematics},
  year      = {1998}
}

Prerequisites

03.02.05
03.02.16
03.09.17

Tier anchors

beginner: Can you rescale a curved space, angle-preservingly, so its curvature is the same everywhere? (informal)
intermediate: Lee-Parker 'The Yamabe Problem' (Bull. AMS 17, 1987) §§1-3; Aubin 'Some Nonlinear Problems in Riemannian Geometry' Ch. 5
master: Lee-Parker 'The Yamabe Problem' (Bull. AMS 17, 1987); Schoen 'Conformal deformation to constant scalar curvature' (J. Diff. Geom. 20, 1984); Aubin 'Nonlinear Analysis on Manifolds' Ch. 5-6

References

Yamabe — On a deformation of Riemannian structures on compact manifolds (1960) · Osaka Math. J. 12, pp. 21--37
Trudinger — Remarks concerning the conformal deformation of Riemannian structures on compact manifolds (1968) · Ann. Scuola Norm. Sup. Pisa 22, pp. 265--274
Aubin — Equations differentielles non lineaires et probleme de Yamabe (1976) · J. Math. Pures Appl. 55, pp. 269--296
Schoen — Conformal deformation of a Riemannian metric to constant scalar curvature (1984) · J. Differential Geom. 20, pp. 479--495
Lee and Parker — The Yamabe Problem (1987) · Bull. Amer. Math. Soc. 17, pp. 37--91
Schoen and Yau — On the proof of the positive mass conjecture in general relativity (1979) · Comm. Math. Phys. 65, pp. 45--76

Estimated time

beginner: 16m
intermediate: 42m
master: 82m