Abresch, Uwe Old and new doubly periodic solutions of the Sinh-Gordon equation. (English) Zbl 0635.35029 New results in nonlinear partial differential equations, Semin. Bonn/FRG 1984, Aspects. Math. E10, 37-73 (1987). [For the entire collection see Zbl 0606.00011.] The author investigates existence of doubly periodic solutions to the equation \(u_{xx}+u_{yy}+\sinh 2u=0\) in connection with a similar problem for a corresponding system of 10 integrable equations of the first order. The results gained are used to the construction of immersed constant mean curvature surfaces in \(R^ 3\). Reviewer: O.Vejvoda Cited in 2 Documents MSC: 35J60 Nonlinear elliptic equations 35B10 Periodic solutions to PDEs 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature Keywords:Sinh-Gordon equation; immersion of surfaces; zero set of solutions; existence; doubly periodic solutions; integrable equations of the first order; constant mean curvature surfaces Citations:Zbl 0606.00011 PDFBibTeX XML