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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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