zbMATH — the first resource for mathematics

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 \(t\).

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.)
Full Text: DOI arXiv
[1] C. Böhm, Manifolds with positive curvature operators are space forms,, Ann. of Math. (2), 167, 1079, (2008) · Zbl 1185.53073
[2] B.-L. Chen, Local pinching estimates in 3-dim Ricci flow,, Math. Res. Lett., 20, 845, (2013) · Zbl 1304.35149
[3] R. S. Hamilton, A compactness property for solutions of the Ricci flow,, Amer. J. Math., 117, 545, (1995) · Zbl 0840.53029
[4] V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds,, Geom. Funct. Anal., 12, 121, (2002) · Zbl 1013.53046
[5] A. Petrunin, Polyhedral approximations of Riemannian manifolds,, Turkish J. Math., 27, 173, (2003) · Zbl 1034.53070
[6] T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications,, J. Reine Angew. Math., 703, 27, (2015) · Zbl 1358.53070
[7] M. Simon, Ricci flow of almost non-negatively curved three manifolds,, J. Reine Angew. Math., 630, 177, (2009) · Zbl 1165.53046
[8] M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below,, J. Reine Angew. Math., 662, 59, (2012) · Zbl 1239.53085
[9] W. Spindeler, <em>\(S^1\)-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature</em>,, Ph.D. Thesis, (2014) · Zbl 1297.53006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.