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.
Used by Vincent Moncrief and Arthur Fischer to study reduced Hamiltonian for Einstein's equations.
be a compact smooth manifold (without boundary) of dimension
assigns to each Riemannian metric
is the volume density associated to the metric
The exponent in the denominator is chosen so that the functional is scale-invariant: for every positive real constant
as measuring the average scalar curvature of
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 define where the infimum is taken over the smooth real-valued functions
): Hölder's inequality implies
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
It follows that if we define where the supremum is taken over all metrics on
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
In the late 1990s, the Yamabe invariant was computed for large classes of 4-manifolds by Claude LeBrun and his collaborators.
In particular, it was shown that most compact complex surfaces have negative, exactly computable Yamabe invariant, and that any Kähler–Einstein metric of negative scalar curvature realizes the Yamabe invariant in dimension 4.
Most of these arguments involve Seiberg–Witten theory, and so are specific to dimension 4.
An important result due to Petean states that if
In light of Perelman's solution of the Poincaré conjecture, it follows that a 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.
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.
holds important topological information.
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.