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Constructing reference metrics on multicube representations of arbitrary manifolds

Lindblom, Lee and Taylor, Nicholas W. and Rinne, Oliver (2016) Constructing reference metrics on multicube representations of arbitrary manifolds. Journal of Computational Physics, 313 . pp. 31-56. ISSN 0021-9991.

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Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss–Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error).

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Additional Information:© 2016 Elsevier Inc. Received 26 November 2014; Received in revised form 13 August 2015; Accepted 10 February 2016; Available online 12 February 2016. We thank Jörg Enders, Gerhard Huisken, James Isenberg, and Klaus Kröncke for helpful discussions about Ricci flow, and Michael Holst and Ralf Kornhuber for helpful discussions on surface finite element methods. L.L. and N.T. thank the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Golm, Germany for their hospitality during a visit when a portion of this research was completed. L.L. and N.T. were supported in part by a grant from the Sherman Fairchild Foundation and by grants DMS-1065438 and PHY-1404569 from the National Science Foundation. O.R. was supported by a Heisenberg Fellowship and grant RI 2246/2 from the German Research Foundation (DFG). We also thank the Center for Computational Mathematics at the University of California at San Diego for providing access to their computer cluster (acquired through NSF DMS/MRI Award 0821816) on which all the numerical tests reported in this paper were performed.
Funding AgencyGrant Number
Sherman Fairchild FoundationUNSPECIFIED
Heisenberg FellowshipUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)RI 2246/2
Subject Keywords:Topological manifolds; Differential structure; Numerical methods; Ricci flow
Record Number:CaltechAUTHORS:20160429-132628179
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Official Citation:Lee Lindblom, Nicholas W. Taylor, Oliver Rinne, Constructing reference metrics on multicube representations of arbitrary manifolds, Journal of Computational Physics, Volume 313, 15 May 2016, Pages 31-56, ISSN 0021-9991, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:66552
Deposited By: Tony Diaz
Deposited On:01 May 2016 17:12
Last Modified:03 Oct 2019 09:57

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