Geodesic
Solutions of the Calapso equation
Serdica Mathematical Journal, Tome 44 (2018) no. 3-4, pp. 341-364.

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In this paper, we introduce a class of surfaces called radial inverse mean curvature surface (RIMC-surfaces) and we show that there is a correspondence between these surfaces and Bryant surfaces in the hyperbolic space H^3 , therefore, the RIMC-surfaces are isothermic. We obtain a Weierstrass type representation for RIMC-surfaces which depends on a meromorphic function and a holomorphic function and we obtain a characterization so that these surfaces are parametrized by lines of curvature. In [3] it is shown that for each isothermic surface parametrized by lines of curvature in the Euclidean space a solution of the Calapso equation is associated, in this work we show that for these surfaces we can associate another solution of the Calapso equation. Moreover, we give explicit solutions of the Calapso equation that depend on holomorphic functions.
Keywords: Lines of curvature, Weierstrass representation, Isothermic surfaces, Calapso equation, 53A30, 32A10, 34M05 
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     title = {Solutions of the {Calapso} equation},
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M. V. Corro, Armando; M. C. Riveros, Karlos; V. Fernandes, Karoline. Solutions of the Calapso equation. Serdica Mathematical Journal, Tome 44 (2018) no. 3-4, pp. 341-364. http://geodesic.mathdoc.fr/item/SMJ2_2018_44_3-4_a3/