Definitions

# Yamabe invariant

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.

## Definition

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

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

where $R_g$ is the scalar curvature of $g$ and $dV_g$ 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 $mathcal\left\{E\right\}\left(g\right)$ 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 $mathcal\left\{E\right\}\left(g\right)$ 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

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

where the infimum is taken over the smooth functions $f$ on $M$. The number $Y\left(g\right)$ 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 $g$, $Y\left(g\right)$ is bounded above by $mathcal\left\{E\right\}\left(g_0\right)$, where $g_0$ is the standard metric on the $n$-sphere $S^n$. The number $mathcal\left\{E\right\}\left(g_0\right)$ is equal to $6\left(2pi^2\right)^\left\{2/3\right\}$ and is often denoted $sigma_1$. It follows that if we define

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

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

## 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

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

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

$sigma\left(M\right) = 4pi chi\left(M\right).$

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

## 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 $CP_2$ 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 $sigma \left(M\right) geq 0$. 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 $sigma_1$ is defined above to be $6\left(2pi^2\right)^\left\{2/3\right\}$.

$M$ $sigma\left(M\right)$ notes
$S^3$ $sigma_1$ the 3-sphere
$S^2 times S^1$ $sigma_1$ the trivial 2-sphere bundle over $S^1$
$S^2 stackrel\left\{sim\right\}\left\{times\right\} S^1$ $sigma_1$ the unique non-orientable 2-sphere bundle over $S^1$
$R mathbb\left\{P\right\}^3$ $sigma_1/2^\left\{2/3\right\}$ computed by Bray and Neves
$R mathbb\left\{P\right\}^2 times S^1$ $sigma_1/2^\left\{2/3\right\}$ 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.

## Topological significance

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

## 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 $mathbb\left\{RP\right\}^3$", Ann. of Math. 159, 407-424 (2004).
• M.J. Gursky and C. LeBrun, "Yamabe invariants and $Spin^c$ 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.
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