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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mn>5</mn> <mo>×</mo> <mn>7</mn> </math></EquationSource> <EquationSource Format="TEX">$5\times 7$</EquationSource> </InlineEquation> quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.</p>

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Compact Bonnet pairs: isometric tori with the same curvatures

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月14日

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  • Alexander I. Bobenko,
  • Tim Hoffmann,
  • Andrew O. Sageman-Furnas

摘要

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 × 7 $5\times 7$ quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.