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

Abstract

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. *

Willmore surfaces in spheres: the DPW approach via the conformal Gauss map

“…In [18] we will prove this conjecture for all compact Riemann surfaces and for the pseudo-Riemannian symmetric space occurring in our Willmore setting.…”

On symmetric Willmore surfaces in spheres II: The orientation reversing case

“…ThenM is the unit disk or C, or S 2 = C ∪ {∞}, together with a complex structure. As in [10], a symmetry of some Willmore surface y : M → S n+2 consists of a pair of maps (µ, S) satisfying y • µ(z) = y(µ(z)) = S(y(z)),…”

“…Finally we obtain that P(C(z, λ −1 ) −1 )P(M (λ) −1 )P(C(µ(z), λ)) is of the form In this case we then obtain f (γ(p)) =Rf (p), whereR denotes the element of O + (1, n + 3) which acts on the light cone containing S n+2 and which represents R after projection to S n+2 . In (4.3), the statement concerning S-Willmore surfaces in Theorem 4.1, page 109 of [10], we stated f (γ(p)) =f (γ(p)) =Rf (p).…”

“…The basic technical reason for this is that the image of the conformal Gauss map is a four-dimensional Lorentzian subspace of R n+4 1 which carries naturally an orientation and is located inside the naturally oriented R n+4 1 and that an orientation reversing symmetry reverses the orientation of the image of the conformal Gauss map. As a consequence we will need to keep track of orientations in addition to the features discussed in [10].…”

Minimal surfaces with non-trivial topology in the three-dimensional Heisenberg group

“…For a detailed proof one can follow the proof of [29,Theorem 31.2], but with Ψi (z) = W + (η i (z) −1 , z, λ) −1 . See for example [24].…”