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Compact Bonnet pairs: isometric tori with the same curvatures
Alexander I. Bobenko1 idref; Tim Hoffmann2 idref; Andrew O. Sageman-Furnas3
1Institute of Mathematics, MA 8-3, Technical University of Berlin, 10623, Berlin, Germanyror
2TUM School of Computation, Information and Technology, Department of Mathematics, Technische Universität München, 85748, Garching, Germanyror
3Department of Mathematics, North Carolina State University, 27695, Raleigh, NC, USAror
Publications Mathématiques de l'IHÉS, Volume 142 (2025), pp. 241-293
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We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. The above approach stems from computational investigations of a $5\times 7$ quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.

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DOI: 10.1007/s10240-025-00159-z

Alexander I. Bobenko  1 ; Tim Hoffmann  2 ; Andrew O. Sageman-Furnas  3

1 Institute of Mathematics, MA 8-3, Technical University of Berlin, 10623, Berlin, Germany ror
2 TUM School of Computation, Information and Technology, Department of Mathematics, Technische Universität München, 85748, Garching, Germany ror
3 Department of Mathematics, North Carolina State University, 27695, Raleigh, NC, USA ror 
Alexander I. Bobenko; Tim Hoffmann; Andrew O. Sageman-Furnas. Compact Bonnet pairs: isometric tori with the same curvatures. Publications Mathématiques de l'IHÉS, Volume 142 (2025), pp. 241-293. doi: 10.1007/s10240-025-00159-z
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