Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method.In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce finite order rotational symmetries. We also prove, for the symmetric space which is the target space of the conformal Gauss map of Willmore surfaces in spheres, the longstanding conjecture of the existence of meromorphic invariant potentials for the conformal Gauss maps of all compact Willmore surfaces in spheres. We also illustrate our results by some concrete examples. *

The family of Willmore immersions from a Riemann surface into S n+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in R n+2 and those which are not conformally equivalent to a minimal surface in R n+2 . On the level of their conformal Gauss maps into Gr 1,3 (R 1,n+3 ) = SO + (1, n + 3)/SO + (1, 3) × SO(n) these two classes of Willmore immersions into S n+2 correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of R 1,n+3 , contains a fixed lightlike vector or where it does not contain such a "constant lightlike vector". Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into S n+2 which are not conformal to a minimal surface in R n+2 . It turns out that our proof also works analogously for minimal immersions into the other space forms.

The conformal geometry of spacelike surfaces in 4-dim Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the known transforms (Darboux transform and spectral transform) of isothermic surfaces by establishing the permutability theorems. Mathematics Subject Classification (2000). Primary 53B25; Secondary 53B30.