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.
It was introduced by Richard Hamilton shortly after the Ricci flow, as an approach to solve the Yamabe problem on manifolds of positive conformal Yamabe invariant.
Recent research
The fixed point solutions for Yamabe flow are described by the Einstein equation, leading to the Einstein manifolds of constant scalar curvature.Yamabe flow considered in works leads to the Einstein-type spaces so that compactness is synonymous with being of Einstein type.[1]
Notes
- ↑ http://arxiv.org/pdf/hep-th/0701084.pdf Towards physically motivated proofs of the Poincar´e and geometrization conjectures