Convergence of the Yamabe Flow for large Energies

We consider the Yamabe or scalar curvature flow on general compact closed manifolds.

By showing convergence of the scalar curvature to its average value in all $L^p$ norms for $t \to \infty$, we deduce via a concentration-compactness arguement that the metrics either converge to a smooth Yamabe metric, or else concentrate in finitely many bubbles. In the presence of at most one bubble we identify a Kazdan-Warner type transversality condition that rules out concentration and therefore implies convergene of the flow.

The condition is very natural and easily verified when the manifold is conformal to the standard sphere.

Using the positive mass Theorem we proof that the criterion also holds on general manifolds of dimensions $3 \le n \le 5$ and in the local conformally flat case.

This is joint work with M. Struwe, Zuerich