Smoothing 3-dimensional polyhedral spaces (Q256348)

Celebrated 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 \(t\).