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.
We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation (ODE) with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mean curvature surface which has a properly immersed end that is asymptotically Delaunay. Furthermore, that end is embedded if the Delaunay surface is unduloidal.
The emergence of large-scale freeform shapes in architecture poses big challenges to the fabrication of such structures. A key problem is the approximation of the design surface by a union of patches, socalled panels, that can be manufactured with a selected technology at reasonable cost, while meeting the design intent and achieving the desired aesthetic quality of panel layout and surface smoothness. The production of curved panels is mostly based on molds.Since the cost of mold fabrication often dominates the panel cost, there is strong incentive to use the same mold for multiple panels. We cast the major practical requirements for architectural surface paneling, including mold reuse, into a global optimization framework that interleaves discrete and continuous optimization steps to minimize production cost while meeting user-specified quality constraints. The search space for optimization is mainly generated through controlled deviation from the design surface and tolerances on positional and normal continuity between neighboring panels. A novel 6-dimensional metric space allows us to quickly compute approximate inter-panel distances, which dramatically improves the performance of the optimization and enables the handling of complex arrangements with thousands of panels. The practical relevance of our system is demonstrated by paneling solutions for real, cutting-edge architectural freeform design projects.
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