Engineering

Curvature

Pure mathematics

Ricci curvature

Riemannian manifold

Gravitational singularity

Stability (learning theory)

Computer science

Limit (mathematics)

Diffeomorphism

Mechanical engineering

Mathematics

Flow (mathematics)

Ricci flow

Mathematical analysis

Geometry

Uniqueness

Conjecture

Manifold (fluid mechanics)

Machine learning

Acta Mathematica

Author(s): Bamler, Richard H; Kleiner, Bruce | Abstract: We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.

Uniqueness and stability of Ricci flow through singularities