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Journal Article

Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity

MPS-Authors
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Jasiulek,  Michael
Astrophysical Relativity, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Korzynski,  Mikolaj
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1111.6523
(Preprint), 772KB

0264-9381_29_15_155010.pdf
(Any fulltext), 1MB

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Citation

Jasiulek, M., & Korzynski, M. (2012). Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity. Classical and quantum gravity, 29(15): 155010. doi:10.1088/0264-9381/29/15/155010.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0012-5E29-2
Abstract
We present a numerical method for solving Weyl's embedding problem which consists of finding a global isometric embedding of a positively curved and positive-definite spherical 2-metric into the Euclidean three space. The method is based on a construction introduced by Weingarten and was used in Nirenberg's proof of Weyl's conjecture. The target embedding results as the endpoint of an embedding flow in R^3 beginning at the unit sphere's embedding. We employ spectral methods to handle functions on the surface and to solve various (non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity range from quasi-local mass and momentum measures to coarse-graining in inhomogeneous cosmological models.