Abstract
We explicitly find the minima as well as the minimum points of the geodesic length functions for the family of filling (hence non-simple) closed curves, \(a^2b^n\) (\(n\ge 3\)), on a complete one-holed hyperbolic torus in its relative Teichmüller space, where a, b are simple closed curves on the one-holed torus which intersect exactly once transversely. This provides concrete examples for the problem to minimize the geodesic length of a fixed filling closed curve on a complete hyperbolic surface of finite type in its relative Teichmüller space.
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Acknowledgements
Part of the work contained in this paper was done during the authors’ visit in the summer of 2021 to the Institute for Advanced Study in Mathematics (IASM) at Zhejiang University, and they thank IASM for its hospitality, and Professor Shicheng Wang for his constant interests and helpful discussions. They would like to thank the anonymous referee for his/her helpful comments that improved the quality of the manuscript. The second author is supported by NSFC grant No. 12171345.
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ZW and YZ wrote the main manuscript text and made the revision, and YZ prepared Fig. 1. All authors reviewed the manuscript.
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Wang, Z., Zhang, Y. Length minima for an infinite family of filling closed curves on a one-holed torus. Geom Dedicata 218, 6 (2024). https://doi.org/10.1007/s10711-023-00856-1
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DOI: https://doi.org/10.1007/s10711-023-00856-1
Keywords
- Filling closed curve
- Relative Teichmüller space
- Minimum of geodesic length function
- Hyperbolic one-holed torus