Symmetry of Embedded Genus-one Helicoids

In this note, we use the Lopez-Ros deformation introduced in [9] to show that any embedded genus-one helicoid must be symmetric with respect to rotation by 180 around a normal line. This partially answers a conjecture of Bobenko from [3]. We also show this symmetry holds for an embedded genus-k helicoid Σ, provided the underlying conformal structure of Σ is hyperelliptic. In [3], Bobenko conjectures that any immersed genus-k helicoid (i.e. a minimally immersed, once punctured genus-k surface with “helicoid-like” behavior at the puncture) is symmetric with respect to rotation by 180 around a line perpendicular to the surface. This conjecture is motivated by the observation in [3] that the period problem for these surfaces is algebraically “well-posed” when there is such a symmetry, but is “over-determined” without it. In this note, we verify Bobenko’s conjecture for embedded genus-one helicoids. That is: Theorem 0.1. Let Σ be an embedded genus-one helicoid. Then there is a line l normal to Σ so that rotation by 180 about l acts as an orientation preserving isometry on Σ. We define a genus-k helicoid to be a complete, minimal surface immersed in R which has genus k, one end, and is asymptotic to a helicoid. A consequence of Theorem 3 of [5] is that any (immersed) minimal surface which is conformally a once-punctured compact genus-k Riemann surface with “helicoid-like” Weierstrass data at the puncture is a genus-k helicoid in this sense. In particular, the above definition encompasses the surfaces studied by Bobenko. Importantly, by Theorem 1.1 of [2], any complete, embedded minimal surface in R with genus k and one end has “helicoid-like” Weierstrass data and hence is a genus-k helicoid. The space of such objects is not vacuous. Weber, Hoffman and Wolf [12] and Hoffman and White [7] have given (very different) constructions of embedded genus-one helicoids – at present it is unknown whether the two constructions give the same surface. Both constructions produce a genus-one helicoid that has, in addition to the orientation preserving symmetry of Theorem 0.1, two orientation reversing symmetries. Whether all genus-one helicoids possess these additional symmetries is also unknown. We emphasize that our argument does not generalize to genus k > 1 because we crucially use the fact that every genus-one Riemann surface admits a large number of biholomorphic involutions – more precisely, that any once-punctured genus-one Riemann surface admits a non-trivial biholomorphic involution. This need not be true for higher genus. Indeed, a priori there may fail to be any nontrivial biholomorphic automorphisms. However, if we restrict attention to genus-k 2000 Mathematics Subject Classification. 53A10. The authors were supported respectively by the NSF grants DMS-0902721 and DMS-0902718.