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 $mathcal\{E\}$ assigns to each Riemannian metric $g$ on $M$ a real number as follows:

: $mathcal\{E\}(g)\; =\; frac\{int\_M\; R\_g\; ,\; dV\_g\}\{left(int\_M\; ,\; dV\_g\; ight)^\{frac\{n-2\}\{n\},$

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 $mathcal\{E\}(g)$ 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 conformalYamabe 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 equalto $6(2pi^2)^\{2/3\}$ and is often denoted $sigma\_1$. 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). Thereal number $sigma(M)$ 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

: $mathcal\{E\}(g)\; =\; int\_M\; R\_g\; ,\; dV\_g\; =\; int\_M\; 2K\_g\; ,\; dV\_g,$

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

: $sigma(M)\; =\; 4pi\; chi(M).$

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 $sigma\; (M)\; geq\; 0$. 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 $sigma\_1$ is defined above tobe $6(2pi^2)^\{2/3\}$.

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 examplesof 3-manifolds for which the invariant is both negative andexactly computable.

Topological significance

The sign of the Yamabe invariant of $M$ holds important topological information. For example, $sigma(M)$ is positiveif and only if $M$ admits a metric of positive scalar curvature [Akutagawa, et al., pg. 73] . The significance of this fact is that much is known about the topology of manifolds with metrics of positive scalar curvature.

ee also

*Yamabe flow
*Yamabe problem
*Obata's theorem

Notes

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 $Spin^c$ 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.