Construction of non-simply connected CMC surfaces via dressing

Unitarization of Loop Group Representations of Fundamental Groups

“…The following proposition (see Theorem 3.5 in [3]) gives a characterization of an important class of unitarizable loop matrices. Recall that an element of SL(2, C) is semi-simple if and only if it is diagonalizable.…”

“…In this paper, we will use v(λ) to denote an eigenvalues of L(λ), opposite to [3], in which v(λ) is used to stand for the imaginary part of an eigenvalues. So, when L(λ) satisfies (2.3), its eigenvalues are v(λ) and 1/v(λ).…”

“…Let r ∈ (0, +∞). We introduce the following twisted r-loop groups (see [4], [2] and [3] for details), in which λ ∈ S 1 r serves as the loop parameter:…”

On the associated family of Delaunay surfaces

“…Henceforth, when we speak of the monodromy representation, or simply monodromy, we tacitly assume that it is induced by an underlying triple (ξ, Φ 0 ,z 0 ) with invariant holomorphic potential γ * ξ = ξ for all γ ∈ ∆. It is shown in [2] that CMC immersions of open Riemann surfaces M can always be generated by such invariant holomorphic potentials.…”

Constant Mean Curvature Surfaces of Any Positive Genus

“…The condition (1.2) is harder to satisfy and makes use of varying the initial condition Φ 0 . These three conditions have been used in a number of papers, starting with the work of Dorfmeister and Haak [3] and by several of the authors while investigating cmc immersions of the n-punctured Riemann sphere, the so called n-Noids [8], [9] and [14]. Another approach to studying embedded cmc 3-Noids can be found in the work of Große-Brauckmann, Kusner and Sullivan [6].…”