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 für Mathematik, Friedrich-Schiller-Universität 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:
[1]

C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2), 167 (2008), 1079-1097. doi: 10.4007/annals.2008.167.1079.

[2]

B.-L. Chen, G. Xu and Z. Zhang, Local pinching estimates in 3-dim Ricci flow, Math. Res. Lett., 20 (2013), 845-855. doi: 10.4310/MRL.2013.v20.n5.a3.

[3]

R. S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math., 117 (1995), 545-572. doi: 10.2307/2375080.

[4]

V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal., 12 (2002), 121-137. doi: 10.1007/s00039-002-8240-1.

[5]

A. Petrunin, Polyhedral approximations of Riemannian manifolds, Turkish J. Math., 27 (2003), 173-187.

[6]

T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications, J. Reine Angew. Math., 703 (2015), 27-41. doi: 10.1515/crelle-2013-0042.

[7]

M. Simon, Ricci flow of almost non-negatively curved three manifolds, J. Reine Angew. Math., 630 (2009), 177-217. doi: 10.1515/CRELLE.2009.038.

[8]

M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below, J. Reine Angew. Math., 662 (2012), 59-94. doi: 10.1515/CRELLE.2011.088.

[9]

W. Spindeler, $S^1$-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature, Ph.D. Thesis, 2014.

show all references

References:
[1]

C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. of Math. (2), 167 (2008), 1079-1097. doi: 10.4007/annals.2008.167.1079.

[2]

B.-L. Chen, G. Xu and Z. Zhang, Local pinching estimates in 3-dim Ricci flow, Math. Res. Lett., 20 (2013), 845-855. doi: 10.4310/MRL.2013.v20.n5.a3.

[3]

R. S. Hamilton, A compactness property for solutions of the Ricci flow, Amer. J. Math., 117 (1995), 545-572. doi: 10.2307/2375080.

[4]

V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal., 12 (2002), 121-137. doi: 10.1007/s00039-002-8240-1.

[5]

A. Petrunin, Polyhedral approximations of Riemannian manifolds, Turkish J. Math., 27 (2003), 173-187.

[6]

T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications, J. Reine Angew. Math., 703 (2015), 27-41. doi: 10.1515/crelle-2013-0042.

[7]

M. Simon, Ricci flow of almost non-negatively curved three manifolds, J. Reine Angew. Math., 630 (2009), 177-217. doi: 10.1515/CRELLE.2009.038.

[8]

M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below, J. Reine Angew. Math., 662 (2012), 59-94. doi: 10.1515/CRELLE.2011.088.

[9]

W. Spindeler, $S^1$-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature, Ph.D. Thesis, 2014.

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