Yamabe Invariant - The Yamabe Invariant in Two Dimensions

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.