Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

Abstract

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.

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

“…Remark 4.2. The proof of this result requires a lengthy argument and will therefore be published in [37]. It is not difficult to verify that f is of finite uniton type.…”

“…This is the main goal of this subsection. We state the main result and refer for a proof (which uses substantially the techniques discussed in the previous sections) to [37].…”

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

“…As to (2) and (3), first we note that there exists no non-orientable translationally equivariant minimal surfaces in R 3 , since the Catenoid is the only translationally equivariant minimal surface in R 3 . Secondly, by Lemma 1.2 of [20] (see also [14]), y is conformally equivalent to a minimal surface in S 3 or H 3 if and only if so is its associated family. As a consequence, the monodromy matrix exp(πL(λ)) takes value in some subgroup of SO + (1, 4) conjugate to SO(4) for all λ if y is conformally equivalent to a minimal surface in S 3 .…”

“…We have for µ(z) = − 1 z and z ∈ C, we see that y is a Willmore immersion (without branched points) from RP 2 to S 4 . Moreover, y = 1 y 0 (y 1 , y 2 , y 3 , y 4 , y 5 ) t is in fact a minimal immersion from RP 2 to S 4 (see [20]) , with induced metric…”

Weierstrass–kenmotsu Representation of Willmore Surfaces in Spheres

“…Moreover, if one uses the loop group formalism to produce all strongly conformally harmonic maps f , one can recognize immediately by looking at the "normalized potential" if the associated harmonic map has a constant light-like vector. It is one of the main results of [36] to show how these exceptional normalized potentials looks like. Excluding this exceptional case, all other normalized potentials yield harmonic maps with frames belonging to the case of the associated Willmore map/maps being nondegenerate.…”