Geometric Flow - Ricci Flow, Willmore Energy, Inverse Mean Curvature Flow, Calabi Flow, Yamabe Flow (Paperback)

Chapters: Ricci Flow, Willmore Energy, Inverse Mean Curvature Flow, Calabi Flow, Yamabe Flow. Source: Wikipedia. Pages: 34. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In differential geometry, the Ricci flow is an intrinsic geometric flow (a process which deforms the metric of a Riemannian manifold) in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric. It plays an important role in Grigori Perelman's solution of the Poincare conjecture; in this context is also called the RicciHamilton flow. Given a Riemannian manifold with metric tensor, we can compute the Ricci tensor, which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called "time" (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation The normalized Ricci flow makes sense for compact manifolds and is given by the equation where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and is the dimension of the manifold. This normalized equation preserves the volume of the metric. The factor of 2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.) Informally, the Ricci flow tends to expand negatively curved regions ...More: http: //booksllc.net/?id=288291