Yamabe invariant

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In mathematics, in the field of differential geometry, the Yamabe invariant (also referred to as the sigma constant) is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe.

1 Definition
2 The Yamabe invariant in two dimensions
3 Examples
4 Topological significance
5 See also
6 Notes
7 References

[edit] Definition

Let $M$ be a compact smooth manifold of dimension $n\geq 2$ . The normalized Einstein-Hilbert functional $\mathcal{E}$ assigns to each Riemannian metric $g$ on $M$ a real number as follows:

$\mathcal{E}(g) = \frac{\int_M R_g \, dV_g}{\left(\int_M \, dV_g\right)^{\frac{n-2}{n}}},$

where $R g$ is the scalar curvature of $g$ and $d V g$ is the volume form associated to the metric $g$ . Note that the exponent in the denominator is chosen so that the functional is scale-invariant. We may think of $\mathcal{E}(g)$ as measuring the average scalar curvature of $g$ over $M$ . It was conjectured by Yamabe that every conformal class of metrics contains a metric of constant scalar curvature (the so-called [Yamabe problem]); it was proven by Yamabe, Trudinger, Aubin, and Schoen that a minimum value of $\mathcal{E}(g)$ is attained in each conformal class of metrics, and in particular this minimum is achieved by a metric of constant scalar curvature. We may thus define

$Y(g) = \inf_{f} \mathcal{E}(e^{2f} g),$

where the infimum is taken over the smooth functions $f$ on $M$ . The number $Y (g)$ is sometimes called the conformal Yamabe energy of $g$ (and is constant on conformal classes).

A comparison argument due to Aubin shows that for any metric $g$ , $Y (g)$ is bounded above by $\mathcal{E}(g_0)$ , where $g 0$ is the standard metric on the $n$ -sphere $S n$ . The number $\mathcal{E}(g_0)$ is equal to $6(2π 2) 2 / 3$ and is often denoted $σ 1$ . It follows that if we define

$\sigma(M) = \sup_{g} Y(g),$

where the supremum is taken over all metrics on $M$ , then $\sigma(M) \leq \sigma_1$ (and is in particular finite). The real number $σ(M)$ is called the Yamabe invariant of $M$ .

[edit] The Yamabe invariant in two dimensions

In the case that $n = 2$ , (so that M is a closed surface) the Einstein-Hilbert functional is given by

$\mathcal{E}(g) = \int_M R_g \, dV_g = \int_M 2K_g \, dV_g,$

where $K g$ is the Gauss curvature of g. However, by the Gauss-Bonnet theorem, the integral of the Gauss curvature is given by $2πχ(M)$ , where $χ(M)$ is the Euler characteristic of M. In particular, this number does not depend on the choice of metric. Therefore, for surfaces, we conclude that

σ(M) = 4πχ(M).

For example, the 2-sphere has Yamabe invariant equal to $8π$ , and the 2-torus has Yamabe invariant equal to zero.

[edit] Examples

In the late 1990s, the Yamabe invariant was computed for large classes of 4-manifolds by LeBrun and his collaborators. In particular, it was shown that most compact complex surfaces have negative, exactly computable Yamabe invariant, and that any Kahler-Einstein metric of negative scalar curvature realizes the Yamabe invariant in dimension 4. It was also shown that the Yamabe invariant of $C P 2$ is realized by the Fubini-Study metric, and so is less than that of the 4-sphere. Most of these arguments involve Seiberg-Witten theory, and so are specific to dimension 4.

An important result due to Petean states that if $M$ is simply connected and has dimension $n \geq 5$ , then $\sigma (M) \geq 0$ . In light of Perelman's solution of the Poincaré conjecture, it follows that a simply connected $n$ -manifold can have negative Yamabe invariant only if $n = 4$ . On the other hand, as has already been indicated, simply connected $4$ -manifolds do in fact often have negative Yamabe invariants.

Below is a table of some smooth manifolds of dimension three with known Yamabe invariant. Recall $σ 1$ is defined above to be $6(2π 2) 2 / 3$ .

$M$	$σ(M)$	notes
$S 3$	$σ 1$	the 3-sphere
$S^2 \times S^1$	$σ 1$	the trivial 2-sphere bundle over $S 1$ ^[1]
$S^2 \stackrel{\sim}{\times} S^1$	$σ 1$	the unique non-orientable 2-sphere bundle over $S 1$
$\R \mathbb{P}^3$	$σ 1 / 2 2 / 3$	computed by Bray and Neves
$\R \mathbb{P}^2 \times S^1$	$σ 1 / 2 2 / 3$	computed by Bray and Neves
$T 3$	$0$	the 3-torus

By an argument due to Anderson, Perelman's results on the Ricci flow imply that the constant-curvature metric on any hyperbolic 3-manifold realizes the Yamabe invariant. This provides us with infinitely many examples of 3-manifolds for which the invariant is both negative and exactly computable.

[edit] Topological significance

The sign of the Yamabe invariant of $M$ holds important topological information. For example, $σ(M)$ is positive if and only if $M$ admits a metric of positive scalar curvature^[2]. The significance of this fact is that much is known about the topology of manifolds with metrics of positive scalar curvature.

[edit] See also

Yamabe flow
Yamabe problem
Obata's theorem

[edit] Notes

^ See Schoen, pg. 135
^ Akutagawa, et al., pg. 73

[edit] References

M.T. Anderson, "Canonical metrics on 3-manifolds and 4-manifolds", Asian J. Math. 10 127--163 (2006).
K. Akutagawa, M. Ishida, and C. LeBrun, "Perelman's invariant, Ricci flow, and the Yamabe invariants of smooth manifolds", Arch. Math. 88, 71-76 (2007).
H. Bray and A. Neves, "Classification of prime 3-manifolds with Yamabe invariant greater than $\mathbb{RP}^3$ ", Ann. of Math. 159, 407-424 (2004).
M.J. Gursky and C. LeBrun, "Yamabe invariants and $S p i n c$ structures", Geom. Funct. Anal. 8965--977 (1998).
O. Kobayashi, "Scalar curvature of a metric with unit volume", Math. Ann. 279, 253-265, 1987.
C. LeBrun, "Four-manifolds without Einstein metrics", Math. Res. Lett. 3 133--147 (1996).
C. LeBrun, "Kodaira dimension and the Yamabe problem," Comm. Anal. Geom. 7 133--156 (1999).
J. Petean, "The Yamabe invariant of simply connected manifolds", J. Reine Angew. Math. 523 225--231 (2000).
R. Schoen, "Variational theory for the total scalar curvature functional for Riemannian metrics and related topics", Topics in calculus of variations, Lect. Notes Math. 1365, Springer, Berlin, 120-154, 1989.