On symmetric Willmore surfaces in spheres I: the orientation preserving case

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.