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Rotational cmc surfaces in terms of Jacobi elliptic functions

  • Denis Polly EMAIL logo
Published/Copyright: October 28, 2025
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Advances in Geometry
From the journal Advances in Geometry Volume 25 Issue 4

Abstract

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten surfaces and can thus be applied to a more general class of surfaces. As another application of this framework, we give explicit parametrizations of a class of rotational constant harmonic mean curvature surfaces in hyperbolic space. In doing so, we close the last gaps in the classification of all channel linear Weingarten surfaces in space forms, started in [24].

Keywords: Constant mean curvature surface; Weingarten surface; space form
MSC 2010: 53A10 (primary); 53C42; 53A35; 53A40; 33E05 (secondary)

Funding statement: This work was done while the author was a JSPS International Research Fellow (Graduate School of Science, Kobe University) and has been supported by the JSPS Grant-in-Aid for JSPS Fellows 22F22701.

Acknowledgements

The author would like to express his gratitude to Udo Hertrich-Jeromin for many years of mentoring and suggesting interesting avenues of research, like this one. Further, the author thanks Joseph Cho, Yuta Ogata, Mason Pember, and Wayne Rossman for many fruitful discussions about this subject.

  1. Communicated by: T. Leistner

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Received: 2024-12-06
Revised: 2025-04-14
Published Online: 2025-10-28
Published in Print: 2025-10-27

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Eigenforms of hyperelliptic curves with many automorphisms
  3. The optimal twisted paper cylinder
  4. Birational rigidity of quartic three-folds with a double point of rank 3
  5. Special divisors on real trigonal curves
  6. The Hoffman–Singleton manifold
  7. Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
  8. Ramification points of homotopies: Enumeration and general theory
  9. On partially ample Ulrich bundles
  10. Rotational cmc surfaces in terms of Jacobi elliptic functions
  11. Locally classical stable planes
  12. Flocks in topological circle planes and their representation in generalised quadrangles
https://doi.org/10.1515/advgeom-2025-0032
Keywords for this article
Constant mean curvature surface; Weingarten surface; space form

Articles in the same Issue

  1. Frontmatter
  2. Eigenforms of hyperelliptic curves with many automorphisms
  3. The optimal twisted paper cylinder
  4. Birational rigidity of quartic three-folds with a double point of rank 3
  5. Special divisors on real trigonal curves
  6. The Hoffman–Singleton manifold
  7. Classification of trees that quasi-inscribe rectangles in the hyperbolic plane
  8. Ramification points of homotopies: Enumeration and general theory
  9. On partially ample Ulrich bundles
  10. Rotational cmc surfaces in terms of Jacobi elliptic functions
  11. Locally classical stable planes
  12. Flocks in topological circle planes and their representation in generalised quadrangles
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