Bonnet pairs and isothermic surfaces

Kamberov, G.; Pedit, Franz; Pinkall, Ulrich

doi:10.1215/s0012-7094-98-09219-5

Cited by 61 publications

(69 citation statements)

References 7 publications

(5 reference statements)

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“…We rely on the treatment of smooth isothermic surfaces by means of quaternionic analysis as developed in [25,26,33]. …”

Section: Smooth Analoguesmentioning

confidence: 99%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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Abstract. We found a class of triangulated surfaces in Euclidean space which have similar properties as isothermic surfaces in Differential Geometry. We call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally. We show that this class is Möbius invariant. Isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes. This definition generalizes isothermic quadrilateral meshes.A consequence is a discrete analog of minimal surfaces. Here the Weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function.

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“…We rely on the treatment of smooth isothermic surfaces by means of quaternionic analysis as developed in [25,26,33]. …”

Section: Smooth Analoguesmentioning

confidence: 99%

Isothermic triangulated surfaces

Lam

Pinkall

2016

Math. Ann.

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“…However, much less is known about Bonnet pairs, which are exactly two noncongruent isometric surfaces with the same mean curvature function. The theory of Bonnet pairs in R 3 is closely related to isothermic surfaces in S 3 [9] and can be studied in the framework of the theory of integrable systems [1]. On the other hand, Lawson and Tribuzy [10] showed that for compact oriented surfaces in R 3 with nonconstant mean curvature, there are at most two surfaces with the given metric and mean curvature.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian Bonnet pairs in $\mathbb CP^2$

2009

Proc. Amer. Math. Soc.

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“…Kamberov, Pedit, and Pinkall have shown that locally each pair of Bonnet mates arises from an isothermic immersion [KPP98]. Thus, if there exist two compact embedded tori that solve Bonnet's problem, then these tori might arise from an isothermic torus.…”

Section: Introductionmentioning

confidence: 99%

Non-special, non-canal isothermic tori with spherical lines of curvature

Bernstein

2000

Trans. Amer. Math. Soc.

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Abstract. This article examines isothermic surfaces smoothly immersed in Möbius space. It finds explicit examples of non-special, non-canal isothermic tori with spherical lines of curvature in two systems by analyzing Darboux transforms of Dupin tori. In addition, it characterizes the property of spherical lines of curvature in terms of differential equations on the Calapso potential of the isothermic immersion, and investigates the effect of classical transformations on this property.

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Bonnet pairs and isothermic surfaces

Cited by 61 publications

References 7 publications

Isothermic triangulated surfaces

Isothermic triangulated surfaces

Lagrangian Bonnet pairs in $\mathbb CP^2$

Non-special, non-canal isothermic tori with spherical lines of curvature

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