The Singly Periodic Genus-one Helicoid

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77–84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: 180◦ rotation about the vertical line; 180◦ rotation about the horizontal lines (the same symmetry); and their composition. Introduction In this paper, we prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in [4] and its significance discussed in [5]. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: 180 rotation about the vertical line; 180 rotation about the horizontal lines (the same symmetry); and their composition—a 180 rotation about a line, orthogonal to the lines on the surface, and passing through a common axis point. This line meets the surface orthogonally and is referred to as a normal symmetry line. The description of the qualitative properties of the surface in the paragraph above is sufficient to determine a two-parameter family of Weierstrass data (1.7) that must contain the Weierstrass data for any surface with these properties—if it exists. One parameter controls the conformal type of the quotient, in this case a rhombic torus. The other can be considered as controlling the placement of the punctures corresponding to the ends. This is worked out in Section 1 and presented in Theorem 1. The proof of existence of the singly periodic genus-one helicoid consists of showing that the period problem ((1.8),(1.9)) is solvable. This is done in Theorem 2 of Section Hoffman was supported by research grant DE-FG03-95ER25250 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy. Hoffman and Wei were supported by research grant DMS-95-96201 of the National Science Foundation, Division of Mathematical Sciences. Research at MSRI is supported in part by NSF grant DMS-90-22140. 1 2 DAVID HOFFMAN, HERMANN KARCHER, AND FUSHENG WEI Figure 1. The periodic genus-one helicoid. 2. In Theorem 3 of Section 3, we prove that the surface is embedded by decomposing a fundamental domain into disjoint graphs. As usual, the existence and embeddedness proofs are independent. We do not use any special properties of the parameters that kill the periods. In fact we show that any singly periodic (by translations) minimal THE SINGLY PERIODIC GENUS-ONE HELICOID 3 surface that, in the quotient, is asymptotic to the helicoid (1.1) and contains a vertical axis and two horizontal parallel lines must be embedded. Other than the helicoid itself, this example was the first embedded minimal surface ever found that is asymptotic to the helicoid. It was one of the important steps in the discovery and construction of the non-periodic genus-one helicoid, whose existence is proved in [5]. We hope that a complete understanding of this periodic surface will be helpful in giving a non-computational proof, which is not complete of this writing, of the embeddedness of the genus-one helicoid. We might have discovered this periodic surface earlier, had we been looking for it at the time. In 1989, the first two authors realized that a construction of Fischer and Koch [1, 2] could be modified to produce singly periodic, embedded minimal surfaces with multiple helicoidal ends. The Fischer-Koch triply periodic surface is formed of pieces congruent to the solution to the disk-type Plateau Problem for the boundary in Figure 2. The surface extends, by 180 rotation about its boundary line segments, to a triply periodic embedded surface. Our modification consisted of two simple steps. First, we realized that the length of the sides marked ai could be increased without limit, producing an embedded minimal graph over a strip. This extends to an embedded, singly periodic surface with six flat ends of Scherk-type. Second, we observed that the fundamental piece could be modified by rotating the horizontal sides a1 and a2 by a fixed angle, say θ. Each θ produces a fundamental embedded piece that extends by 180 rotation to an embedded minimal surface, asymptotic to three coaxial helicoids, and invariant under a vertical screw motion of the form p → ep+ (0, 0, 8b). Its Weierstrass representation is suggestive of the surface to which we now turn our attention. 1. Determination of the Weierstrass Representation The helicoid can be described by the data g = z, dh = i dz z (1.1) on S = C − {0} in the Weierstrass representation X(p) = X(p0) + Re ∫ p