Smoothing 3-dimensional polyhedral spaces.

*(English)*Zbl 1346.53040Celebrated theorems by A. D. Aleksandrov and Yu. G. Reshetnyak imply that the metric of any general (possibly non-regular) convex surface in \(\mathbb{R} ^{3}\) can be approximated by a sequence of \(2\)-dimensional Riemannian manifolds with controlled integral curvatures. The authors study the problem of approximation of a \(3\)-dimensional polyhedral manifold of curvature \(\geq0\) in the sense of A. D. Aleksandrov by Riemannian manifolds of non-negative curvature. The main theorem states that given a compact \(3\)-dimensional polyhedral manifold \(P\) with non-negative Aleksandrov’s curvature, there is a Ricci flow \(L^{t}\) (on a \(3\)-dimensional manifold) converging to \(P\) as \(t\rightarrow0^+\) such that the sectional curvature of \(L^{t}\) is non-negative for every sufficiently small positive \(t\).

Reviewer: Igor G. Nikolaev (Urbana)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |

52B70 | Polyhedral manifolds |

53C70 | Direct methods (\(G\)-spaces of Busemann, etc.) |

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