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Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1689–1709 | Cite as

Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature

  • Svenja HüningEmail author
  • Johannes Wallner
Open Access
Article
  • 103 Downloads

Abstract

This paper studies well-definedness and convergence of subdivision schemes which operate on Riemannian manifolds with nonpositive sectional curvature. These schemes are constructed from linear ones by replacing affine averages by the Riemannian centre of mass. In contrast to previous work, we consider schemes without any sign restriction on the mask, and our results apply to all input data. We also analyse the Hölder continuity of the resulting limit curves. Our main result states that if the norm of the derived scheme (resp. iterated derived scheme) is smaller than the corresponding dilation factor then the adapted scheme converges. In this way, we establish that convergence of a linear subdivision scheme is almost equivalent to convergence of its nonlinear manifold counterpart.

Keywords

Refinement processes Riemannian geometry Geodesic averaging Hölder continuity 

Mathematics Subject Classification (2010)

41A25 65D05 65D17 

Notes

Acknowledgments

Open access funding provided by Austrian Science Fund (FWF).

Funding information

The authors acknowledge the support of the Austrian Science Fund (FWF): This research was supported by the doctoral programme Discrete Mathematics (grant no. W1230) and by the SFB-Transregio programme Discretization in geometry and dynamics (grant no. I705).

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Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut f. GeometrieTU GrazGrazAustria

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