Yamabe Invariant - Definition

Definition

Let be a compact smooth manifold of dimension . The normalized Einstein–Hilbert functional assigns to each Riemannian metric on 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 .