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The elliptic Sinh Gordon equation and the construction of toroidal soap bubbles. (English) Zbl 0697.35044

Calculus of variations and partial differential equations, Proc. Conf., Trento/Italy 1986, Lect. Notes Math. 1340, 275-301 (1988).
[For the entire collection see Zbl 0641.00013.]
H. Wente showed in 1984 that there exist immersed tori of constant mean curvature in \({\mathbb{R}}^ 3\). The idea is to start with doubly-periodic solutions u of the equation (*) \(\Delta u+\lambda \sinh u=0\) in \({\mathbb{R}}^ 2\). The function u can then be used to construct an immersion F(x,y) but the problem is to show that F is doubly periodic, i.e. that the surface F “closes up”. Wente showed that for suitable periodicity parameters and suitable \(\lambda\) this is indeed the case.
In this paper the author investigates all possible solutions u of the equation (*) which for some rectangle R satisfy: \(u=0\) on \(\partial R\), \(u\geq 0\) on R. For \(0<\lambda <\lambda_ 1(R)\) he succeeds in proving that there exists a unique nontrivial solution which can be described in terms of classical elliptic functions. For \(\lambda_ k\downarrow 0\) the solutions \(u_ k\) tend to \(-2 \log | g(z)|^ 2\) where g(z) is the symmetric conformal map of R onto the unit disk (Theorem 2 and Theorem 1). The proof of Theorem 1 uses a priori bounds that are derived in section 2. There is also a section which contains the discussion of an iterative procedure that is useful in computing solutions.
Reviewer: M.Grüter

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
33E05 Elliptic functions and integrals
30C20 Conformal mappings of special domains

Citations:

Zbl 0641.00013