Abstract. We examine the space of surfaces in R 3 which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space Mk of all such surfaces with k ends (where surfaces are identified if they differ by an isometry of R 3) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has no L 2 −nullspace we prove that Mk is locally the quotient of a real analytic manifold of dimension 3k − 6 by a finite group (i.e. a real analytic orbifold), for k ≥ 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension of Mk is independent of the topology of the underlying punctured Riemann surface to which Σ is conformally equivalent. These results also apply to hypersurfaces of H n+1 with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded. I.

Abstract. Complete, conformally flat metrics of constant positive scalar curvature on the complement of k points in the n-sphere, k ≥ 2, n ≥ 3, were constructed by R. Schoen [S2]. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension k. For a generic set of nearby conformal classes the moduli space is shown to be a k−dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds. I.

this paper we shall discuss hypersurfaces M of space forms of constant curvature; where curvature means one of the symmetric functions of curvature associated to the second fundamental form. The values of the constant will be chosen so that the linearized equation will be an elliptic equation on M . For example, for surfaces in

These notes are about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure of the smooth surfaces using conserved quantities, which can be converted into a discrete theory in a natural way. About referencing: We do not attempt to give a complete reference list, and omit what is already referenced in [59]. We list only articles referenced in the body of the text, or that were written after [59] was published, or were otherwise not included in the reference list in [59], or that were referenced in [59] but need to be updated. About using quaternions: In following with the historical development of the field, we use a model that involves quaternions. However, the use of a more standard model has some advantages, as it can be applied in more general dimensions and settings (see Chapter 10 here, for example), and sometimes gives less cluttered computations. It would be a good exercise to convert this text into one involving a more standard quaternion-free model, but we do not do that here (see [27]), and instead only make occasional comments about this. Acknowledgements: Primary thanks must go to Udo Hertrich-Jeromin, who carefully and patiently taught the author more than half of the material in this text. The author also owes thanks to many others for numerous mathematical tips: Fran

Introduction A circle C in R 3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genus-one immersed constant mean curvature surfaces with boundary C that are approximated by compact domains in Wente tori; however, this has not been proved. Still one has the conjecture: Conjecture 1 A compact constant mean curvature surface bounded by a circle is a spherical cap if either of the following conditions hold: 1. The surface has genus 0 and is immersed; 2. The surface is embedded. If M is a compact embedded constant mean curvature surface in R 3 with boundary C<F14

Abstract. We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms R 3, S 3 and H 3. Additionally, we compute the extended frame for any associated family of Delaunay surfaces. 1. Introduction. Surfaces that minimize area under a volume constraint have constant mean curvature (cmc). The generalized Weierstraß representation [3] for non-minimal cmc surfaces involves solving a holomorphic complex linear 2 × 2 system of ordinary differential equations (ode) on a Riemann surface with values in a loop group. A

Dedicated to Professor Takeshi Sasaki on his sixtieth birthday Abstract. We give necessary and sufficient local conditions for the simultaneous unitarizability of a set of analytic matrix maps from an analytic 1-manifold into SLnC under conjugation by a single analytic matrix map. We apply this result to the monodromy arising from an integrable partial differential equation to construct a family of k-noids, genus-zero constant mean curvature surfaces with three or more ends in Euclidean, spherical and hyperbolic 3-spaces.

Abstract. We compute the flux of Killing fields through ends of constant mean curvature 1 in hyperbolic space, and we prove a result conjectured by Rossman, Umehara and Yamada: the flux matrix they have defined is equivalent to the flux of Killing fields. We next give a geometric description of embedded ends of finite total curvature. In particular, we show that we can define an axis for these ends that are asymptotic to a catenoid cousin. We also compute the flux of Killing fields through these ends, and we deduce some geometric properties and some analogies with minimal surfaces in Euclidean space. 1.

Abstract. In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space. 1.

Abstract. 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 nontrivial 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. Introduction. The Gauß map of a constant mean curvature (cmc) surface in three dimensional Euclidean space R 3 is a harmonic map [32], which can be obtained as a projection of a horizontal holomorphic map from the universal cover of the surface into a certain loop group [42]. Based on this loop group approach, there is a generalized Weierstrass type representation for cmc surfaces due to Dorfmeister, Pedit and Wu [11]. It involves solving a