Cmc Surfaces In

This paper presents a unified treatment of constant mean curvature (CMC) surfaces in the simply-connected 3-dimensional space forms R 3 , S 3 and H 3 in terms of meromorphic loop Lie algebra valued 1-forms. We discuss global issues such as period problems and asymptotic behaviour involved in the construction of CMC surfaces with non-trivial topology. We prove existence of new examples of complete non-simply-connected CMC surfaces in all three space forms, with a special emphasis on the case where the surface is homeomorphic to a thrice punctured sphere. We explicitly compute the extended frame for any associated family of Delaunay surfaces and prove two general asymptotics results which are applied to our new examples. Introduction. The Gauß map of a constant mean curvature (CMC) surface in three dimensional Euclidean space R 3 is a harmonic map [28] and such a harmonic map can be obtained as a projection of a horizontal holomorphic map from the universal cover of the surface into a certain loop group [36]. There is a Weierstrass type representation for simply-connected CMC surfaces, commonly called the DPW method [11]. It involves solving a holomorphic complex linear 2 × 2 system of ordinary differential equations (ODE) with values in a loop group, and subsequently Iwasawa decomposing the solution into two factors, one of which is the moving frame of the surface. Even though in general both of these steps are computationally not explicit, and thus obstructive to keeping track of the topology, the DPW method has been successfully implemented in software [29] and has provided new insights in recent years and enabled the discovery of new examples of non-simply-connected CMC surfaces [10], [19], [20]. In particular, the construction of CMC immersions with the topology of the n-punctured Riemann sphere, the so called n-noids, via the DPW method allows us to present new techniques in solving period problems and describing asymptotics. The problem in constructing surfaces with non-trivial topology lies in manipulating the ODE by varying both the coefficients as well as the initial condition. There remain many open questions as to how the ODE affects the geometry of the resulting surface since the metric of the resulting surface is buried deep within this procedure. On the other hand, this method allows full control over the Hopf differential, thus yielding information of the location and order of umbilic points, which occur naturally on compact CMC surfaces of genus …