Scalar Curvature and Topology

Scalar curvature is the weakest of the curvatures: one number per point, the average of all sectional curvatures (9A.4 Curvature and What It Means). Negative scalar curvature says almost nothing about a manifold, since every manifold of dimension at least three carries a metric with negative scalar curvature. Positive scalar curvature is different. It is obstructed: the 3-torus, for example, carries no metric with $R > 0$. And it can be built: it survives connected sums, through thin necks.

This chapter surveys what is known, at the level needed later. Two reasons make it more than a detour. The Ricci flow preserves positive scalar curvature, and in three dimensions the flow with surgery classifies exactly which closed manifolds carry it. That classification is a consequence of Perelman's work, proved in 12C.1 Reading Off the Topology, and it is stated here only as a target. The second reason is physical. In general relativity, scalar curvature is energy density, and the positive mass theorem turns "$R \geq 0$" into "the total mass is nonnegative".

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

• explain why negative scalar curvature is unobstructed and positive scalar curvature is not;
• state the Schoen–Yau and Gromov–Lawson results on the torus and on surgery;
• describe the Gromov–Lawson neck and check its positive scalar curvature in a rotationally symmetric model;
• state the classification of closed 3-manifolds with positive scalar curvature, and its status;
• state the positive mass theorem and compute the scalar curvature of the Schwarzschild metric.

6.1The positive mass theorem

6.2Negative scalar curvature is cheap

Jerry Kazdan and Frank Warner showed in 1975 that on a closed manifold of dimension $n \geq 3$, every smooth function that is negative somewhere is the scalar curvature of some metric. In particular every closed manifold of dimension at least three has a metric of constant negative scalar curvature. Joachim Lohkamp went further in 1994: every manifold of dimension at least three carries a metric of negative Ricci curvature. So negative curvature conditions of the averaged kinds carry no topological information. Contrast sectional curvature, where $K < 0$ forces the universal cover to be $\mathbb{R}^n$ (Cartan–Hadamard, 9A.7 Jacobi Fields and Curvature versus Topology).

6.3Positive scalar curvature is obstructed

Three lines of argument obstruct positive scalar curvature ($R > 0$, "PSC").

• Spinors (Lichnerowicz, 1963). On a closed spin manifold, a Bochner-type formula for the Dirac operator, $D^2 = \nabla^*\nabla + \frac R4$, shows that PSC forbids harmonic spinors, and by the index theorem a topological invariant, the $\hat A$-genus, must vanish. Nigel Hitchin (1974) refined the invariant.
• Minimal hypersurfaces (Schoen–Yau, 1979). The stability argument of 9A.8 Submanifolds and Minimal Surfaces: in a 3-manifold with PSC, every stable minimal surface is a sphere. In $T^3$, minimising area in the homology class of a 2-torus produces a stable minimal torus, a contradiction. So $T^3$ has no metric with $R > 0$. Schoen and Yau extended the argument by induction on dimension, slicing by minimal hypersurfaces.
• Enlargeability (Gromov–Lawson, 1980–83). Using spinors on covering spaces, Mikhail Gromov and Blaine Lawson proved that no torus $T^n$, and more generally no closed spin manifold that admits a map of nonzero degree to a torus, carries PSC.

Since $T^n$ has a flat metric, the dividing line falls exactly at zero: flat is possible, positive is not. That a torus cannot do better than flat was conjectured by Robert Geroch, and the two proofs above settled it.

6.4Positive scalar curvature survives surgery

A connected sum removes a small ball from each manifold and joins the boundary spheres with a tube $S^{n-1}\times[0, 1]$. The idea of the proof is that a thin tube has huge positive scalar curvature, $(n - 1)(n - 2)/\varepsilon^2$ for a round $S^{n-1}(\varepsilon)$ factor, from its small spheres. The bending needed to attach it can be done with negative curvature contributions that stay smaller. In a rotationally symmetric model you can check this directly (Figure 6.1, Exercise 6.5). Codimension at least $3$ is needed so that the attached tube has spheres of dimension at least $2$, whose curvature is positive.

6.5Three dimensions: the classification

Combining these facts with Perelman's work gives a complete answer in dimension three.

The "if" direction is the surgery theorem: spherical space forms are positively curved, $S^2\times S^1$ has $R = 2$ with the product metric, and connected sums preserve PSC. The "only if" direction needs the Ricci flow with surgery. PSC is preserved by the flow, since $\partial_tR \geq \Delta R + \frac2nR^2$, and it forces the flow to become extinct in finite time (Exercise 6.7). The surgeries cut the manifold along necks into the pieces that are listed. This is proved only in 12C.1 Reading Off the Topology, after the whole of Perelman's argument. Here it is a target, and an example of the kind of topological conclusion the flow delivers.

6.6History

Lichnerowicz's vanishing theorem appeared in 1963. Kazdan and Warner's results date from 1975, Lohkamp's from 1994. Schoen and Yau's work on minimal surfaces, scalar curvature and the positive mass theorem appeared in 1979, Gromov and Lawson's surgery theorem in 1980 and their work on enlargeable manifolds in 1983. Witten's proof of the positive mass theorem appeared in 1981. Huisken–Ilmanen and Bray proved the Riemannian Penrose inequality in 2001. The three-dimensional classification followed from Perelman's 2002–03 papers.

6.7Exercises

Solution

$g = e^{2w}|dx|^2$ with $w = 2\log u$. With $n = 3$ and flat background, $R_g = e^{-2w}(-4\Delta w - 2|\nabla w|^2)$. Here $\nabla w = \frac{2\nabla u}{u}$ and $\Delta w = \frac{2\Delta u}{u} - \frac{2|\nabla u|^2}{u^2}$, so $-4\Delta w - 2|\nabla w|^2 = -\frac{8\Delta u}{u}$, and $R_g = -8u^{-5}\Delta u$. The function $1 + \frac{m}{2|x|}$ is harmonic away from $0$.

Solution

With $r = |x|$ and $r' = \frac{m^2}{4r}$, the conformal factor transforms as $(1 + \frac{m}{2r'})^4|dx'|^2 = (1 + \frac{2r}{m})^4\frac{m^4}{16r^4}|dx|^2 = (\frac{m}{2r} + 1)^4|dx|^2$, using $|dx'| = \frac{m^2}{4r^2}|dx|$ for an inversion. The fixed set of an isometric reflection is totally geodesic (9A.8 Submanifolds and Minimal Surfaces). At $r = \frac m2$, $u = 2$, so the area is $4\pi\frac{m^2}{4}\cdot16 = 16\pi m^2$, and $\sqrt{A/16\pi} = m$.

Solution

(a) $-\frac{4\varphi''}{\varphi} = -\frac{1 - \varphi'^2}{\varphi^2}$, and adding $\frac{2(1 - \varphi'^2)}{\varphi^2}$ gives the claim. (b) $\varphi'' = -\frac{\varphi}{\rho^2}$ and $\varphi'^2 = \cos^2\frac{s_0 - s}{\rho}$, so $1 - \varphi'^2 = \frac{\varphi^2}{\rho^2}$ and $R = \frac{4}{\rho^2} + \frac{2}{\rho^2}$. (c) At fixed $\varphi$ and $\varphi'$, $R$ is an affine function of $\varphi''$, so it takes values between its values at the endpoints of any interval of $\varphi''$.

Solution

$T^3$: no (Schoen–Yau, Gromov–Lawson). $S^2\times S^1$: yes. The Poincaré sphere is $S^3/\Gamma$ with $\Gamma$ the binary icosahedral group (7A.9 The Poincaré Conjecture, Precisely): yes. $\mathbb{RP}^3\#\mathbb{RP}^3$: yes, a connected sum of space forms. A hyperbolic manifold is aspherical with infinite fundamental group and is prime, not on the list: no. $L(5, 1)\#(S^2\times S^1)$: yes.

Solution

(a) At a point where $R(\cdot, t)$ attains its minimum, $\Delta R \geq 0$, so $\partial_tR \geq \frac2nR^2 = \frac2n\rho^2$ there; Hamilton's trick (6A.4 Maximum Principles) turns this into the differential inequality for $\rho$. (b) Compare with the solution of $y' = \frac2ny^2$, $y(0) = \rho(0)$, which is $\frac{\rho(0)}{1 - \frac2n\rho(0)t}$ and blows up at $\frac{n}{2\rho(0)}$; $R$ is finite while the flow exists. (c) $\frac{n}{2n(n - 1)} = \frac{1}{2(n - 1)}$: the bound is sharp for the sphere, where $R$ is constant and $|\operatorname{Ric}|^2 = \frac{R^2}{n}$ exactly.