Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

Abstract

A theorem on the unitarizability of loop group valued monodromy representations is presented and applied 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, the extended frame for any associated family of Delaunay surfaces is computed.We identify Euclidean three-space R 3 with the matrix Lie algebra su 2 . The double cover of the isometry group under this identification is SU 2 su 2 . Let T denote the stabilizer of ∈ su 2 under the adjoint action of SU 2 on su 2 . We shall view the two-sphere as S 2 = SU 2 /T. Lemma 1. The mean curvature H of a conformal immersion f : M → su 2 is given byProof. Let U ⊂ M be an open simply connected set with coordinate z : U → C. Writing df = f z dz and df = fz dz, conformality is equivalent to f z , f z = fz, fz = 0 and the existence of a function v ∈ C ∞ (U, R + ) such that 2 f z , fz = v 2 . Let N : U → SU 2 /T be the Gauss map with lift F : U → SU 2 such that N = F F −1 and df = vF ( − dz + + dz)F −1 . The mean curvature is H = 2v −2 f zz , N and the Hopf differential is Q dz 2 with Q = f zz , N . Hence [df ∧ df ] = 2iv 2 N dz ∧ dz. Then F −1 dF = (1/2v)((−v 2 Hdz − 2Qdz)i − + (2Qdz + v 2 Hdz)i + − (v z dz − vzdz)i ). This allows us to compute d * df = iv 2 HN dz ∧ dz and proves the claim.

Unitarization of Loop Group Representations of Fundamental Groups

“…We note that in [9], a result very similar to Theorem 5.1 is proved by Nick Schmitt using a quite different method.…”

Minimal n -noids in hyperbolic and anti-de Sitter 3-space

“…1. In the case of S 3 theorem 4.3 reconstructs a two-dimensional subfamily of the three-dimensional family of trinoids constructed in [30] without the isosceles constraint.…”

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].…”

“…In this case, Lemma 5.1 is then clearly true, so without loss of generality we may assume that [M 0 , g −1 ] = 0. Thus we can apply the gluing theorem [14] (see also [8]) to conclude there exists an initial condition h such that the monodromy group of hΦ is unitary. Example 1.…”

“…By the gluing theorem [14], the immersion f via the Sym-Bobenko formula [2], in (5.5) is defined when using r-Iwasawa splitting [10] for r < 1 and r sufficiently close to 1. Then, up to a rigid motion and homothety of R 3 , we find numerically that f has the following properties (see Figure 2):…”