The Yamabe Problem and Critical Exponents

6C.2 Critical Points and the Mountain Pass found that the mountain pass method fails at the critical exponent $p = \frac{n+2}{n-2}$, because bubbles concentrate and Palais–Smale sequences lose compactness, and that Pohozaev's identity forbids solutions on star-shaped domains. 6C.3 Symmetry, Noether’s Theorem and Moving Planes classified the bubbles on all of $\mathbb{R}^n$: they are the round metric of the sphere, seen through stereographic projection. This chapter shows that the critical exponent is not an analyst's curiosity but the natural exponent of conformal geometry, and follows the problem that made it famous, the Yamabe problem, from Hidehiko Yamabe's flawed proof of 1960 to Richard Schoen's solution of 1984. Along the way it develops the general method for variational problems at a critical exponent: compare the energy level with the energy of a bubble. Below the bubble's energy, minimising sequences cannot afford to concentrate, and compactness returns.

The other half of the chapter explains why Perelman never had to face this difficulty. His entropy $\mathcal W$, minimised in the definition of $\mu(g, \tau)$ (12A.3 The 𝓦-Entropy), is built on the logarithmic Sobolev inequality, whose nonlinearity $w^2\log w^2$ grows more slowly than any power above $2$. It is subcritical, so on a closed manifold minimisers always exist (Oscar Rothaus, 1981), and Perelman could write "it is not hard to show that in the definition of $\mu$ there always exists a smooth minimizer". The contrast between the two problems, one at the edge of compactness and one safely inside it, is one reason the Ricci flow proof works so smoothly at this point: the analysis of the entropy never meets bubbles.

By the end of this chapter you will be able to:

• derive the transformation law for scalar curvature under a conformal change $\tilde g = u^{4/(n-2)}g$, and recognise the critical exponent in it;
• define the Yamabe functional and invariant, compute the invariant of the sphere, and relate it to the sharp Sobolev constant;
• prove Aubin's inequality $Y(M, [g]) \leq Y(S^n)$ by concentrating test functions, and explain why strict inequality gives a minimiser;
• describe Aubin's and Schoen's verification of strict inequality, and the role of the positive mass theorem;
• state Lions's concentration compactness and Struwe's global compactness theorem, and explain the Brezis–Nirenberg problem;
• prove that Perelman's $\mathcal W$-functional, written in $w = (4\pi\tau)^{-n/4}e^{-f/2}$, is coercive on $H^1$ and attains its minimum, and explain why the logarithm is subcritical;
• relate the Yamabe flow to the Ricci flow.

4.1Scalar curvature under conformal change

Let $g$ be a Riemannian metric (on an open set of $\mathbb{R}^n$, or on a closed manifold, 6B.10 Harmonic Coordinates and the Regularity of Metrics) and $\tilde g = e^{2\varphi}g$ a conformal metric: angles are unchanged, lengths are rescaled by $e^\varphi$ point by point. The scalar curvatures are related by

a formula derived in Book 9A (9A.5 Computing Curvature); in dimension two it is $\tilde K = e^{-2\varphi}(K - \Delta\varphi)$, the Gauss curvature equation of 5A.5 Uniformization and the Two-Dimensional Ricci Flow. The gradient term is quadratic, but it disappears after a change of unknown. In dimension $n \geq 3$ write $e^{2\varphi} = u^{4/(n-2)}$ with $u > 0$. Then (Exercise 4.5)

The operator $L_g = -a_n\Delta + R$ is the conformal Laplacian, and the equation for a conformal metric of constant scalar curvature $\tilde R \equiv c$ is

The critical Sobolev exponent is forced by geometry. The power $\frac{n+2}{n-2}$ appears because $u^{4/(n-2)}$ is the conformal factor and $u^{-(n+2)/(n-2)}$ makes the formula covariant: $L_g$ transforms by $L_{\tilde g}(v) = u^{-\frac{n+2}{n-2}}L_g(uv)$, and the volume form is $dV_{\tilde g} = u^{\frac{2n}{n-2}}dV_g$, with $\frac{2n}{n-2} = 2^*$ the Sobolev exponent of 4A.11 Sobolev Embeddings and Critical Exponents.

The Yamabe problem. Hidehiko Yamabe asked in 1960 whether every metric on a closed manifold of dimension $n \geq 3$ is conformal to one of constant scalar curvature. In dimension two the answer is the uniformisation theorem (5A.5 Uniformization and the Two-Dimensional Ricci Flow). In higher dimensions the conformal class of a metric is a huge set (one function's worth of metrics), much smaller than the space of all metrics, and constant scalar curvature is one equation, so the count is right; but the equation sits exactly at the critical exponent.

4.2The Yamabe functional

Solutions are critical points of the Yamabe functional

For $u > 0$, $Q_g(u)$ is the total scalar curvature of $\tilde g = u^{4/(n-2)}g$ divided by $\operatorname{Vol}(\tilde g)^{(n-2)/n}$, a scale-invariant quantity that depends only on $\tilde g$: by the transformation law and an integration by parts, $\int\tilde R\,dV_{\tilde g} = \int uL_gu\,dV_g$. The Euler–Lagrange equation of $Q_g$ is the constant scalar curvature equation, with $c$ determined by the value of $Q$. The Yamabe invariant of the conformal class is

and its sign is the sign of the first eigenvalue of $L_g$ (Kazdan and Warner; 6B.16 Eigenvalue Problems and Perelman’s λ): negative, zero or positive according as the conformal class contains a metric of negative, zero or positive scalar curvature.

The sphere. On the unit sphere $S^n$ the infimum is attained by constants, and by every function whose conformal metric is round: the conformal transformations of $S^n$ (Möbius transformations) act on the minimisers, so the set of minimisers is noncompact. Through stereographic projection these minimisers are exactly the bubbles of 6C.3 Symmetry, Noether’s Theorem and Moving Planes, and

with $\omega_n$ the volume of the unit $n$-sphere ($R = n(n - 1)$, and $Q(1) = n(n-1)\omega_n/\omega_n^{(n-2)/n}$). On $\mathbb{R}^n$, where $R = 0$, the same number is the best constant in the sharp Sobolev inequality of Thierry Aubin and Giorgio Talenti (1976):

with equality exactly for the bubbles. This is stated here; the proof is by symmetric rearrangement, which reduces it to radial functions, and an ODE computation (or by optimal transport, 6C.11 Optimal Transport and Ricci Curvature).

Why the direct method fails, and when it does not. For $1 \leq q < \frac{n+2}{n-2}$, the subcritical problem of minimising $\int(a_n|\nabla u|^2 + Ru^2)$ subject to $\int|u|^{q+1} = 1$ has a minimiser by the direct method, because $H^1 \hookrightarrow L^{q+1}$ is compact (Rellich, 4A.11 Sobolev Embeddings and Critical Exponents). At the critical exponent the embedding $H^1 \hookrightarrow L^{2^*}$ is continuous but not compact: a minimising sequence can concentrate at a point, where after rescaling it looks like a bubble on $\mathbb{R}^n$, and its weak limit can be zero. The key observation is that concentration has a price.

So $Y(S^n)$ is the energy of concentration, and any minimising sequence that concentrates at a point has energy at least $Y(S^n)$ in the limit. If the infimum is strictly below, minimising sequences cannot concentrate.

The Yamabe problem is thereby reduced to an inequality between numbers: show $Y(M, [g]) < Y(S^n)$ unless $(M, g)$ is conformal to the round sphere. Yamabe's 1960 paper had claimed existence in general; Neil Trudinger found the gap in 1968 and proved existence when $Y(M, [g]) \leq 0$, or small; Aubin (1976) proved the strict inequality when $n \geq 6$ and $M$ is not locally conformally flat, by refining the test functions $u_\varepsilon$ so that the Weyl curvature at $p$ produces a negative correction of order $\varepsilon^4|W(p)|^2$; and Schoen (1984) proved it in the remaining cases ($n = 3, 4, 5$, or $M$ locally conformally flat).

Schoen's step and the positive mass theorem. In the remaining cases the local geometry at a point is too weak to lower the energy, and Schoen used the global geometry instead. The Green's function $G_p$ of the conformal Laplacian with pole at $p$ defines a metric $G_p^{4/(n-2)}g$ on $M\setminus\{p\}$ that is scalar flat and asymptotically flat: near $p$, which has become infinity, it looks like Euclidean space outside a ball. Such a metric has an ADM mass, the coefficient of the leading correction to flatness at infinity, a quantity first defined in general relativity (by Arnowitt, Deser and Misner, 1962) as the total energy of an isolated gravitating system. Schoen built test functions by gluing a bubble at $p$ into the Green's function, and found that $Q$ falls below $Y(S^n)$ by an amount proportional to the mass. The positive mass theorem of Schoen and Shing-Tung Yau (1979; Witten gave a spinor proof in 1981), that an asymptotically flat metric of nonnegative scalar curvature has nonnegative mass, with mass zero only for Euclidean space, then gives the strict inequality unless $(M, g)$ is conformally the round sphere. A statement about the energy of gravitating systems completed a problem in conformal geometry.

4.3Concentration compactness and bubbles

The Yamabe problem is the model for every variational problem at a critical exponent, and the analysis of its failures of compactness has been made general.

So compactness can fail only by concentration at points, and each concentration costs a definite quantum of energy, at least that of a bubble. Michael Struwe's global compactness theorem (1984) completes the picture for Palais–Smale sequences of $-\Delta u = |u|^{2^*-2}u + \lambda u$ on a bounded domain: after passing to a subsequence, $u_k$ is, up to an error tending to zero in $H^1$, the sum of a solution $u_0$ and finitely many rescaled bubbles, and its energy is the sum of their energies. This bubble decomposition reappears for harmonic maps and minimal surfaces in 6C.8 Monotonicity, ε-Regularity and Tangent Cones, where Sacks–Uhlenbeck bubbling (6C.7 Harmonic Maps and Their Heat Flow) is the two-dimensional version.

The Brezis–Nirenberg problem. On a bounded domain $U \subset \mathbb{R}^n$, Pohozaev forbids positive solutions of $-\Delta u = u^{2^*-1}$ if $U$ is star-shaped (6C.2 Critical Points and the Mountain Pass). Haïm Brezis and Louis Nirenberg (1983) showed that a small linear perturbation restores existence: for $n \geq 4$ and $0 < \lambda < \lambda_1(U)$,

has a solution. The proof is the method of this chapter in its purest form: the term $\lambda u$ lowers the energy of a concentrating bubble below the threshold $\frac1nS^{n/2}$ (by a test-function computation like Aubin's), and below the threshold Palais–Smale sequences are compact. In dimension three the lowering is weaker, and for the ball the solutions exist exactly for $\frac14\lambda_1 < \lambda < \lambda_1$: a dimension-dependent threshold of the same kind as Schoen's low-dimensional cases.

4.4The Yamabe flow

Richard Hamilton proposed in 1989 to solve the Yamabe problem by a parabolic flow within the conformal class,

where $\bar R$ is the average of the scalar curvature, so that volume is preserved. Writing $g = u^{4/(n-2)}g_0$, it is a nonlinear heat equation for $u$ with the critical nonlinearity. The flow exists for all time (Hamilton; Ye, 1994); its convergence to a constant scalar curvature metric was proved by Rugang Ye (1994) for locally conformally flat manifolds, and in general by Hartmut Schwetlick and Michael Struwe (2003) and Simon Brendle (2005, 2007), with the positive mass theorem again entering in low dimensions. In dimension two, the Yamabe flow is the Ricci flow ($\operatorname{Ric} = \frac R2g$), and Hamilton's proof that the normalised Ricci flow on a surface converges to constant curvature (11A.12 Ricci Flow on Surfaces) is the two-dimensional Yamabe flow theorem. In higher dimensions the two flows diverge: the Yamabe flow changes only the conformal factor and prescribes the scalar curvature, which is one equation, while the Ricci flow moves the whole metric and controls the full Ricci tensor, $\frac{n(n+1)}{2}$ equations. The Yamabe flow is the closer analogue of the scalar-curvature problems, the Ricci flow the one that can change topology-detecting geometry.

4.5Why Perelman's minimisers always exist

Perelman's entropy, for a closed manifold $(M, g)$ and a scale $\tau > 0$, is

and $\mu(g, \tau)$ is its infimum over $f$ with $\int(4\pi\tau)^{-n/2}e^{-f}\,dV = 1$ (12A.3 The 𝓦-Entropy). On $\mathbb{R}^n$ with the flat metric, $\mathcal W \geq 0$ is the Gaussian logarithmic Sobolev inequality (6A.12 Entropy, Information and Diffusion). Substitute

Then $|\nabla f|^2(4\pi\tau)^{-n/2}e^{-f} = 4|\nabla w|^2$ and $f = -\log w^2 - \frac n2\log(4\pi\tau)$, so (Exercise 4.8)

The nonlinearity is $w^2\log w^2$, which sits between $w^2$ and any power $w^{2+\delta}$: for large $w$ it grows faster than $w^2$, which is why the inequality is stronger than a Poincaré inequality, but more slowly than $w^{2+\delta}$ for every $\delta > 0$, which is why it is subcritical. The following is the existence half of Rothaus's theorem.

Compare with the Yamabe problem. There the nonlinearity $u^{2^*}$ scales exactly like the gradient term, so a concentrating bubble costs a fixed energy and can be a minimising sequence's escape route; here $w^2\log w^2$ is dominated by the gradient term at small scales, so concentrating costs unboundedly much entropy ($\mathcal W \to +\infty$ as a minimising candidate concentrates, since $\tau\int|\nabla w|^2$ grows like the inverse square of the concentration scale while the logarithm grows only like its logarithm), and there is no escape. Perelman's $\mu$ is a well-behaved function of $(g, \tau)$, with smooth minimisers to differentiate along the flow; his proof of no local collapsing (12A.4 κ-Noncollapsing) uses exactly the opposite direction: on a collapsed region, a well-chosen test function makes $\mathcal W$ very negative, contradicting the lower bound for $\mu$ that monotonicity propagates from the initial metric.

Two limits of $\mu$ complete the picture, both stated by Perelman in his §3.1. As $\tau \to 0$, the minimiser concentrates at the scale $\sqrt\tau$ where the manifold looks Euclidean, and $\mu(g, \tau) \to 0$, the Euclidean value; this is the log-Sobolev inequality of $\mathbb{R}^n$ seen in the tangent space. As $\tau \to \infty$, if $\lambda(g) > 0$, then $\mu(g, \tau) \to +\infty$: the term $\tau\int(4|\nabla w|^2 + Rw^2) \geq \tau\lambda(g)$ dominates (6B.16 Eigenvalue Problems and Perelman’s λ).

4.6History

Yamabe's paper appeared posthumously in 1960. Trudinger found its gap and proved partial results in 1968. Aubin's paper on the Yamabe problem and Aubin's and Talenti's sharp Sobolev constants appeared in 1976. Schoen and Yau proved the positive mass theorem in 1979 and Witten gave his proof in 1981; Arnowitt, Deser and Misner had defined the mass in 1962. Rothaus proved existence of log-Sobolev minimisers on compact manifolds in 1981. Brezis and Nirenberg's paper appeared in 1983; Schoen completed the Yamabe problem in 1984, the same year as Struwe's global compactness theorem and the first of Lions's concentration-compactness papers. Lee and Parker's survey appeared in 1987. Hamilton introduced the Yamabe flow in 1989; Ye (1994), Schwetlick and Struwe (2003) and Brendle (2005, 2007) proved its convergence. Perelman's $\mu$ appeared in 2002.

4.7Exercises

Solution

$\nabla\varphi = \frac{2}{n-2}\frac{\nabla u}{u}$, so $|\nabla\varphi|^2 = \frac{4}{(n-2)^2}\frac{|\nabla u|^2}{u^2}$ and $\Delta\varphi = \frac{2}{n-2}\big(\frac{\Delta u}{u} - \frac{|\nabla u|^2}{u^2}\big)$. Then $2(n - 1)\Delta\varphi + (n - 1)(n - 2)|\nabla\varphi|^2 = \frac{4(n - 1)}{n - 2}\frac{\Delta u}{u} - \frac{4(n - 1)}{n - 2}\frac{|\nabla u|^2}{u^2} + \frac{4(n - 1)}{n - 2}\frac{|\nabla u|^2}{u^2} = a_n\frac{\Delta u}{u}$. So $\tilde R = u^{-4/(n-2)}\big(R - a_n\frac{\Delta u}{u}\big) = u^{-\frac{4}{n-2} - 1}(Ru - a_n\Delta u)$, and $\frac{4}{n-2} + 1 = \frac{n+2}{n-2}$.

Solution

(a) $R = n(n - 1)$, $\nabla1 = 0$, so the numerator is $n(n - 1)\omega_n$ and the denominator $\omega_n^{(n-2)/n}$; the ratio is $n(n - 1)\omega_n^{2/n}$. (b) $6(2\pi^2)^{2/3} \approx 6\cdot(19.74)^{2/3} \approx 6\cdot7.30 \approx 43.8$. (c) $a_nK_n^{-2} = \frac{4(n-1)}{n-2}\cdot\frac{n(n-2)\omega_n^{2/n}}{4} = n(n-1)\omega_n^{2/n}$.

Solution

Substituting $x = \varepsilon y$: $\nabla U_\varepsilon(x) = \varepsilon^{-n/2}(\nabla U)(y)$, so $\int|\nabla U_\varepsilon|^2dx = \varepsilon^{-n}\varepsilon^n\int|\nabla U|^2dy$; $U_\varepsilon^{2^*} = \varepsilon^{-n}U^{2^*}(y)$, so $\int U_\varepsilon^{2^*} = \int U^{2^*}$. But $\int U_\varepsilon^2\,dx = \varepsilon^{-(n-2)}\varepsilon^n\int U^2\,dy = \varepsilon^2\int U^2$, finite when $U^2 \sim |y|^{-2(n-2)}$ is integrable at infinity, that is $2(n - 2) > n$, $n \geq 5$. So the curvature term is $O(\varepsilon^2)$ while the main terms are fixed: concentrating bubbles see only the flat tangent space. (For $n = 3, 4$ one must cut $U_\varepsilon$ off, and the term is $O(\varepsilon)$ or $O(\varepsilon^2|\log\varepsilon|)$.)

Solution

$w^2 = (4\pi\tau)^{-n/2}e^{-f}$, so $\log w^2 = -f - \frac n2\log(4\pi\tau)$, that is $f = -\log w^2 - \frac n2\log(4\pi\tau)$. Also $\nabla w = -\frac12w\nabla f$, so $|\nabla f|^2w^2 = 4|\nabla w|^2$. Then $\int[\tau(|\nabla f|^2 + R) + f - n]w^2 = \int[\tau(4|\nabla w|^2 + Rw^2)] + \int fw^2 - n$, and $\int fw^2 = -\int w^2\log w^2 - \frac n2\log(4\pi\tau)$.

Solution

$\int|\nabla w_\varepsilon|^2 = \varepsilon^{-2}\int|\nabla\phi|^2$. And $w_\varepsilon^2 = \varepsilon^{-n}\phi^2(x/\varepsilon)$, so $\int w_\varepsilon^2\log w_\varepsilon^2 = \int\phi^2(y)\big(\log\phi^2(y) - n\log\varepsilon\big)dy = \int\phi^2\log\phi^2 + n\log\frac1\varepsilon$. So the combination is $4\tau\varepsilon^{-2}\int|\nabla\phi|^2 - n\log\frac1\varepsilon - \int\phi^2\log\phi^2 \to +\infty$: the gradient term grows like $\varepsilon^{-2}$, the entropy term only like $\log\frac1\varepsilon$. For the Yamabe quotient both terms scale the same way and concentration costs a fixed amount; here it costs an unbounded amount, which is why minimising sequences of $\mathcal W$ stay compact.

Solution

(a) Write $\tau\int(4|\nabla w|^2 + Rw^2) = (1 - \theta)\tau\int(\ldots) + \theta\tau\int(\ldots)$; bound the first part below by $(1 - \theta)\tau\lambda$ and the second by $\theta\tau(4\|\nabla w\|^2 + \min R)$. (b) With the choice of $\theta$, $(1 - \theta)\tau\lambda + \theta\tau\min R \geq \frac12(1 - \theta)\tau\lambda =: c\tau$. For $4\theta\tau \geq 1$, $4\theta\tau\|\nabla w\|^2 - \int w^2\log w^2 \geq \|\nabla w\|^2 - \|\nabla w\|^2 - C_1 = -C_1$. So $\mathcal W \geq c\tau - C_1 - n - \frac n2\log(4\pi\tau)$ for every admissible $w$, hence for $\mu$; the linear term beats the logarithm. (c) As $\tau \to 0$ the gradient term $4\tau\|\nabla w\|^2$ has a vanishing coefficient, and absorbing $\int w^2\log w^2$ into it needs a constant $C_{4\tau}$ that grows like $\frac n2\log\frac1\tau$ (by the scaling computation of Exercise 4.9), which nearly cancels $-\frac n2\log(4\pi\tau)$. Deciding the limit needs Perelman's rescaling argument: at scale $\sqrt\tau$ the manifold is nearly Euclidean, and the sharp Euclidean inequality $\mathcal W \geq 0$ gives $\mu \to 0$.