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I was wondering if someone knows which is the parameterization that we made in these cases. I calculated till $$Z(s)=\left(\frac{1}{2iH}(1-e^{-2iHs})+C\right)e^{2iHs},$$ but can't figure out what parameterization was done after that and why we have the following values ​​for $y(s)$ and $x^{'}(s)$!

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user981656
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Feb 21 '22 at 20:06
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    Why a differential geometry tag? Is the definition of a Kenmotsu manifold remotely relevant? – Ted Shifrin Feb 21 '22 at 20:07
  • Are you by chance asking about the paper K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tôhoku Math. J. 32 (1980), 147--153...? – Andrew D. Hwang Feb 22 '22 at 15:04
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    @AndrewD.Hwang , yes ! As guidance I have a book called :"Surfaces with Constant Mean Curvature K. Kenmotsu ,pages 39-42" I need to determine where are the results coming from. the results are written quite directly and I don't understand, and I have to write them step by step. And I got stuck in the case where H is a constant(other than 0) because I don't understand what parameterization was chosen to write the final result. – user981656 Feb 22 '22 at 17:13
  • I don't have any special insight beyond the calculations in Kenmotsu's paper. As is commonly the case, $(x(s), y(s))$ is an arclength parametrization of the profile curve; he defines $Z(s) = y(s)(y'(s) + ix'(s))$, shows $Z' -2iHZ - 1 = 0$, and deduces the expressions after two pages of calculation with judicious choices of auxiliary function. – Andrew D. Hwang Feb 22 '22 at 18:52

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