Yamabe flow

In differential geometry, the Yamabe flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold.Yamabe flow is for noncompact manifolds. It is the negative L²-gradient flow of the (normalized) total scalar curvature, restricted to a given conformal class: it can be interpreted as deforming a Riemannian metric to a conformal metric of constant scalar curvature, when this flow converges.

The Yamabe flow was introduced in response to Richard S. Hamilton's work on the Ricci flow and Rick Schoen's solution of the Yamabe problem on manifolds of positive conformal Yamabe invariant.

Main results

The fixed points of the Yamabe flow are metrics of constant scalar curvature in the given conformal class. The flow was first considered by Bennett Chow,^[1] working only in the conformally flat case, basically following the lead of Hamilton's previous work on the Ricci flow. A reproof of the Yamabe conjecture using the Yamabe flow in arbitrary conformal classes on compact manifolds was then given by Ye.^[2] A more definitive theory of the Yamabe flow on compact manifolds was later given by Brendle,^[3] who showed that, at least in low dimensions, the flow converges quite generally, although the limiting constant-scalar-curvature metic is typically no longer a Yamabe minimizer in this context. By contrast, the flow is less well understood on complete, non-compact manifolds, and there are still open problems in this context which remain topics of current research.

Notes

1. Bennett Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math. 45 (1992), no. 8, 1003–1014.
2. Rugang Ye,, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994), no. 1, 35–50.
3. Simon Brendle, Convergence of the Yamabe flow for arbitrary initial energy.J. Differential Geom. 69 (2005), no. 2, 217–278.