Harmonic tori in De Sitter spaces $$S^{2n}_1$$ S 1 2 n

Abstract

We show that all superconformal harmonic immersions from genus one surfaces into de Sitter spaces S 2n 1 with globally defined harmonic sequence are of finite-type and hence result merely from solving a pair of ordinary differential equations. As an application, we prove that all Willmore tori in S 3 without umbilic points can be constructed in this simple way.

Toda frames, harmonic maps and extended Dynkin diagrams

“…Define ∇ λ = d + ad ϕ λ and note that (7) says precisely that ∇ λ is a flat connection in the trivial bundle C/Λ × g C . With X as above we have ∇ ′ λ Y = 0.…”

“…A straightforward computation shows that these vector fields commute and so finding solutions to (12) is merely a matter of solving a pair of commuting ordinary differential equations. This yields a rather special class of solutions to the Maurer-Cartan equations (7) and hence of harmonic maps to symmetric spaces and primitive maps to k-symmetric spaces, k > 2. The flows of X, Y are easily seen to evolve on spheres in Ω d .…”

“…Some solutions to (7) can be obtained simply by solving a pair of commuting ordinary differential equations on a finite-dimensional loop algebra. These unusually simple solutions are said to be of finite type.…”

“…Classical solutions of sigma-models are given by harmonic maps into (non-compact) as pseudo-Riemannian manifolds. In [7] we study an explicit example, namely harmonic tori in de Sitter spaces S m 1 . In particular we apply the theory of this paper to the superconformal such maps with globally defined harmonic sequence to see that they may all be obtained by integrating a pair of commuting vector fields on a finite-dimensional vector space.…”

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

“…As a matter of fact, a local parametrization of all minimal submanifolds f : M m → H n with index of relative nullity ν = m − 2 was given in [12] in terms of certain elliptic space-like surfaces in either the Lorentzian sphere or the Lorentzian flat space according to n − m being even or odd, respectively. Moreover, from the results in [3], [9] and [11] it is clear that this parametrization can be used to construct complete examples of any dimension other than generalized cones. Let U ⊂ M m be an open subset where the index of relative nullity ν = s > 0 is constant.…”