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Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications

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Abstract

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be included in our construction. Then we present two applications of our main results: the generalized Green’s theorem and the uniqueness of signature for planar Jordan curves with finite \(p\)-variation for \(1\leqslant p<2\).

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Acknowledgments

The authors wish to thank Professor Terry Lyons for his valuable suggestions on the present paper. The authors are supported by the Oxford-Man Institute at University of Oxford. The first author is also supported by ERC (Grant Agreement No.291244 Esig).

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Authors and Affiliations

  1. The Oxford-Man Institute, Eagle House, Walton Well Road, Oxford, OX2 6ED, UK

    H. Boedihardjo & X. Geng

  2. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    X. Geng

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  1. H. Boedihardjo
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  2. X. Geng
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Correspondence to X. Geng.

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Communicated by G. Kerkyacharian.

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Boedihardjo, H., Geng, X. Simple Piecewise Geodesic Interpolation of Simple and Jordan Curves with Applications. Constr Approx 42, 161–180 (2015). https://doi.org/10.1007/s00365-014-9257-z

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  • DOI: https://doi.org/10.1007/s00365-014-9257-z

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