, 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
be a compact
smooth manifold of dimension
. The normalized Einstein-Hilbert functional
assigns to each Riemannian metric
a real number as follows:
where is the scalar curvature of and is the volume form associated to the metric . Note that the exponent in the denominator is chosen so that the functional is scale-invariant. We may think of as measuring the average scalar curvature of over . 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 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
where the infimum is taken over the smooth functions on . The number is sometimes called the conformal
Yamabe energy of (and is constant on conformal classes).
A comparison argument due to Aubin shows that for any metric , is bounded above by , where
is the standard metric on the -sphere . The number is equal
to and is often denoted . It follows that if we define
where the supremum is taken over all metrics on , then (and is in particular finite). The
real number is called the Yamabe invariant of .
The Yamabe invariant in two dimensions
In the case that
, (so that M
is a closed surface
) the Einstein-Hilbert functional is given by
where is the Gauss curvature of g. However, by the Gauss-Bonnet theorem, the integral of the Gauss curvature is given by
, where is the Euler characteristic of M. In particular, this number does not depend on the choice of metric. Therefore, for surfaces, we conclude that
For example, the 2-sphere has Yamabe invariant equal to , and the 2-torus has Yamabe invariant equal to zero.
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
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 is simply connected and has dimension , then . In light of Perelman's solution of the Poincaré conjecture, it follows that a simply connected -manifold can have negative Yamabe invariant only if . On the other hand, as has already been indicated, simply connected -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 is defined above to
|| notes |
|| the 3-sphere |
|| the trivial 2-sphere bundle over |
|| the unique non-orientable 2-sphere bundle over |
|| computed by Bray and Neves |
|| computed by Bray and Neves |
|| 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
The sign of the Yamabe invariant of
holds important topological information. For example,
if and only if
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.
- 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 ", Ann. of Math. 159, 407-424 (2004).
- M.J. Gursky and C. LeBrun, "Yamabe invariants and 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.