Totally Real Minimal Tori In

In this paper we show that all totally real superconformal minimal tori in CP 2 correspond with doubly-periodic finite gap solutions of the Tzitzéica equation ωzz̄ = e −2ω − eω. Using the results on the Tzitzéica equation in integrable system theory, we describe explicitly all these tori by Prym-theta functions. Introduction Over the past few years the integrable system approach played an important role in the theory of minimal surfaces and harmonic maps. One of the most classical and striking results is on the tori with constant mean curvature (CMC) in R, which correspond the doubly-periodic solutions to the sinh-Gordon equation. Firstly let us recall the history of the research on CMC tori since it contains all the seeds useful to our case. In 1987, Wente’s significant existence theorem [30] provoked Abresch [1] to classify all CMC tori having one family of planar curvature lines. These surfaces are given explicitly in terms of elliptic functions by reducing the PDE to ODEs solvable. The approach to seek the general solution to the problem has developed into a major area of research. The main points of view are due to Pinkall and Sterling [26] et al, whose approach can be expressed in terms of Hamiltonian systems and Loop groups, Bobenko [3], who sees it in terms of the much studied finite-gap solutions for the sinh-Gordon equation and the corresponding Baker-Akhiezer function, and Hitchin [18] whose approach is influenced by twistor theory. Later on, Bolton-Pedit-Woodward ([5]) showed that the existence of the correspondence between minimal surfaces in CP n and the solutions of affine Toda equations for SU(n + 1). And they further proved that any superconformal harmonic 2-torus in CP n is of finite type. They posed that together with the results in [14] this accounts for all (conformal) harmonic 2-tori in CP . Unfortunately, even in the case of CP , the Clifford torus x(z) = [e, e, e 2z−ǭ2z̄], ǫ = e 2πi 3 , which is an isometric embedding of flat torus whose corresponding lattice in C is generated by 2π/3 and 2πi, is the only example of superconformal harmonic tori in CP 2 known to us during a long period. On the other hand, there has been much attention given to totally real (Lagrangian) submanifolds in CP n in the classical differential geometry community. Then a natural problem was posed to give an explicit construction of totally real minimal tori in CP . Moreover, the explicit 1991 Mathematics Subject Classification. Primary 53C42, Secondary 53C43, 53D12.