Construction of higher genus minimal surfaces with one end and finite total curvature

We prove that there exist complete minimal surfaces in the Euclidean 3-space with one Enneper-type end and finite total curvature which have two parameters 7, k and are of genus jk, where j and k are positive integers. Our main problem is the period problem', each surface has j periods to be killed. We prove that these periods can be killed simultaneously. Introduction. Recently, many minimal surfaces with higher genus and finite total curvature have been found. Costa [Co] found a complete minimal surface of genus one in R. Hoffman and Meeks [HM2] proved its embeddedness, and they found higher genus embedded surfaces which are similar to Costa's surface, but with higher-order rotational symmetry. Wohlgemuth [W] proved rigorously the existence of several higher genus minimal surfaces which have embedded ends, including a surface which was found by Hoίfman, Meeks and Callahan only numerically. In 1982, Chen-Gackstatter [CG] found surfaces which have one end and are of genera one and two. The genus one C-G surface was generalized by Karcher [K] and the genus two C-G surface was generalized by Thayer [T]. These generalizations are similar to C-G surfaces, but with higher winding order at the end. Since minimal surfaces in R are given by Weierstrass data and path integrals, we always must check well-definedness of the surfaces, and this is called the period problem. Chen-Gackstatter [CG] also gave Weierstrass data for a genus three surface, but they did not solve its period problem. Thayer [T] conjectured that the period problem can be solved for arbitrary genus and gave numerical evidence to support this. Espίrito-Santo [E] solved the genus three case with a numerical argument. Rossman suggested to me a homotopy argument (which can be thought of as an intermediate value theorem of several variables) for solving the period problem. Wohlgemuth [W] used the homotopy argument to solve period problems for minimal surfaces with four embedded ends. In this paper, we solve the period problem of the generalized C-G surfaces for arbitrary genus (2.1) by using the intermediate value theorem of several variables. 1991 Mathematics Subject Classification. Primary 53A10; Secondary 53C42. 230 K. SATO MAIN THEOREM. There exist one-ended complete minimal surfaces X(Mjtk) in R 3 of genus jk with total curvature — 4(y + \)kπ for all j,k=l,2,. In Section 1, we summarize some basic fact about minimal surface theory. In Section 2, we introduce the generalized C-G surfaces X(Mjtk) and discuss their symmetries. In Section 3, we show that each X(Mjtk) is complete and regular in /? , and we see that the symmetries in Section 2 can be used to reduce the number of periods fromyfc toy. Furthermore, we write down a concrete condition for solving the periods. In Section 4, we rewrite some of the results of Chen-Gackstatter [CG] and Thayer [T] in preparation for Section 5. In Section 5, we solve the period problem inductively, by assuming several inequalities. Finally in Section 6, we prove these inequalities. REMARK. Once we prove the existence of certain value for the parameters a2, . , aj9 ceR so that the minimal immersion (2.1) is well-defined, we say that the period problem is solved. Though we could not show uniqueness for the value of a2, . , aj9 c, our numerical computations suggest that the surfaces are unique. (Recently, in the case j= 2 and fc=l, Lopez, Martin and Rodriguez [LMR] have shown the uniqueness for the value a2 and c.) More information about the numerical results and several conjectures are written in [T]. Thanks are also due to the referee and Dr. Rossman for valuable suggestions. 1. Basic properties. The following Theorems 1.1-1.3 were given by Osserman [Osl][Os2] (see also [HM1]). THEOREM 1.1. Let M be a Riemann surface, g: M->Cu {00} a meromorphie function and dh a holomorphίc \-form on M. We define a vector-valued \-form Φ by (1.1) * = (Φι, Φ2, Φι) = (~-gdh, i— + ̂ \dh, 2dh. Then the real part (1.2) PQ defines a minimal mapping into /? , which is well-defined on M if and only if