Abstract
We introduce a family of variational functionals for spinor fields on a compact Riemann surface M that can be used to find close-to-conformal immersions of M into \(\mathbb {R}^3\) in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on M.
Keywords
- Isometric and conformal immersions
- Variational problems
- Non-linear Dirac equation
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Acknowledgements
Authors supported by SFB Transregio 109 “Discretization in Geometry and Dynamics” at Technical University Berlin. Third author partially supported by an RTF grant from the University of Massachusetts Amherst. Fifth author partially supported by the Einstein Foundation. Software support for images provided by SideFX. We thank Stefan Sechelmann for the abstract hyperbolic triangulated surface used in Fig. 1.
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Chern, A., Knöppel, F., Pedit, F., Pinkall, U., Schröder, P. (2021). Finding Conformal and Isometric Immersions of Surfaces. In: Hoffmann, T., Kilian, M., Leschke, K., Martin, F. (eds) Minimal Surfaces: Integrable Systems and Visualisation. m:iv m:iv m:iv m:iv 2017 2018 2018 2019. Springer Proceedings in Mathematics & Statistics, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-030-68541-6_2
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