gu Surface uniformization means that all metric surfaces can be conformally mapped to one of the three canonical domains, the sphere, the plane and the hyperbolic space

Surface uniformization means that all metric surfaces can be conformally mapped to one of the three canonical domains, the sphere, the plane and the hyperbolic space. The figures show the uniformization for surfaces with various topologies. Unformization converts general 3D geometric problems to 2D problems in these canonical domains. Ricci flow is a powerful geometric analytic tool, which has been applied to prove Poincare conjecture. Ricci flow is a parabolic system of partial differential equations which acts like the heat equation to spread the curvature of a Riemannian metric evenly over the surface to produce a metric of constant curvature. Computational discrete Ricci flow is the practical method to compute surface uniformization, it has many important applications in many engineering fields.

http://www3.cs.stonybrook.edu/~gu/index.html#algorithm


http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=5972&context=rtd


Drop shapes and internal flow patterns by numerical solution of the Navier-Stokes equations