Like heat flows from hot to cold and makes uniform temperature, curvature can flow to make constant curvature {Ricci flow equation}. However, Ricci flow allows singularities, with different starting geometries. For example, dumbbell shape tends to make two spheres with point between, rather than one large sphere. Thin rod tends to have singular point at one end {cigar singularity}. If sphere replaces singularity, Ricci flow can continue. Ricci flow can find possible shapes in compact spaces (Richard Hamilton) [1982].
Grigory Perelman [2003] used Ricci flow to show that all Ricci-flow-procedure singularity types can morph into spheres or tubes in finite time, so procedures can remove them from space.
Mathematical Sciences>Topology>Surface
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Date Modified: 2022.0224