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Smoothing 3-dimensional polyhedral spaces. (English) Zbl 1346.53040
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 \(t\).

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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