# Hyperelliptic Curves that Generate Constant Mean Curvature Tori in ℝ3

• N. M. Ercolani
• H. Knörrer
• E. Trubowitz
Chapter
Part of the Progress in Mathematics book series (PM, volume 115)

## Abstract

Let u be a solution of the elliptic-sinh Gordon equation
$${u_{w\bar w}} + \sinh u = 0$$
(1)
on the simply connected domain Ω ⊂ ℂ. There is an algorithm that associates an immersion F of Ω in ℝ3 to u with constant mean curvature ½ (see e.g. [3]). To implement ite first solves
$${\psi _w} = - \frac{1}{2}\left( {\begin{array}{*{20}{c}} {{u_w}} \\ i \end{array}\begin{array}{*{20}{c}} i \\ { - {u_w}} \end{array}} \right)\psi$$
(1)
$${\psi _\varpi } = - \frac{1}{{2.1}}\left( {\begin{array}{*{20}{c}} 0 \\ {{e^u}} \end{array}\begin{array}{*{20}{c}} {{e^{ - u}}} \\ 0 \end{array}} \right)\psi$$
(1)
for
$$\psi = \left( {\begin{array}{*{20}{c}} {{\psi _1}} \\ {{\psi _2}} \end{array}\begin{array}{*{20}{c}} {(w)} \\ {(w)} \end{array}} \right)$$
on Ω. Equation (1) is the consistency condition for (2) and (3).

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### References

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