English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In 1883, Darboux gave a local classification of isothermic surfaces with onefamily of planar curvature lines using complex analytic methods. His choice ofreal reduction cannot contain tori. We classify isothermic tori with one familyof planar curvature lines. They are found in the second real reduction ofDarboux's description. We give explicit theta function formulas for the familyof plane curves. These curves are particular area constrained hyperbolicelastica. With a Euclidean gauge, the Euler--Lagrange equation is lower orderthan expected. In our companion paper (arXiv:2110.06335) we use such isothermictori to construct the first examples of compact Bonnet pairs: two isometrictori related by a mean curvature preserving isometry. They are also the firstpair of isometric compact immersions that are analytic. Additionally, we studythe finite dimensional moduli space characterizing when the second family ofcurvature lines is spherical. Isothermic tori with planar and sphericalcurvature lines are natural generalizations of Wente constant mean curvaturetori, discovered in 1986. Wente tori are recovered in a limit case of ourformulas.

Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica