# American Institute of Mathematical Sciences

2015, 22: 12-19. doi: 10.3934/era.2015.22.12

## Smoothing 3-dimensional polyhedral spaces

 1 Steklov Institute, St. Petersburg, Russian Federation 2 Institut für Mathematik, Friedrich-Schiller-Universität Jena, Germany 3 Mathematics Department, Pennsylvania State University, United States 4 National Research University, Higher School of Economics, Moscow, Russian Federation

Received  November 2014 Published  June 2015

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
Citation: Nina Lebedeva, Vladimir Matveev, Anton Petrunin, Vsevolod Shevchishin. Smoothing 3-dimensional polyhedral spaces. Electronic Research Announcements, 2015, 22: 12-19. doi: 10.3934/era.2015.22.12
##### References:
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show all references

##### References:
 [1] C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms,, \emph{Ann. of Math. (2)}, 167 (2008), 1079.  doi: 10.4007/annals.2008.167.1079.  Google Scholar [2] B.-L. Chen, G. Xu and Z. Zhang, Local pinching estimates in 3-dim Ricci flow,, \emph{Math. Res. Lett.}, 20 (2013), 845.  doi: 10.4310/MRL.2013.v20.n5.a3.  Google Scholar [3] R. S. Hamilton, A compactness property for solutions of the Ricci flow,, \emph{Amer. J. Math.}, 117 (1995), 545.  doi: 10.2307/2375080.  Google Scholar [4] V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds,, \emph{Geom. Funct. Anal.}, 12 (2002), 121.  doi: 10.1007/s00039-002-8240-1.  Google Scholar [5] A. Petrunin, Polyhedral approximations of Riemannian manifolds,, \emph{Turkish J. Math.}, 27 (2003), 173.   Google Scholar [6] T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications,, \emph{J. Reine Angew. Math.}, 703 (2015), 27.  doi: 10.1515/crelle-2013-0042.  Google Scholar [7] M. Simon, Ricci flow of almost non-negatively curved three manifolds,, \emph{J. Reine Angew. Math.}, 630 (2009), 177.  doi: 10.1515/CRELLE.2009.038.  Google Scholar [8] M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below,, \emph{J. Reine Angew. Math.}, 662 (2012), 59.  doi: 10.1515/CRELLE.2011.088.  Google Scholar [9] W. Spindeler, $S^1$-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature,, Ph.D. Thesis, (2014).   Google Scholar
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