2001/10/09 by Rafe Mazzeo, Frank Pacard, Mazzeo, Rafe +3
Mathematics · #53A10 #53C40 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53A10 #msc:53C40
paper · pdf · doi:10.48550/arxiv.math/0110099
32 pages
arxiv created 2001/10/09 · openalex publication_date 2001/10/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We first prove a general gluing theorem which creates new nondegenerate constant mean curvature surfaces by attaching half Delaunay surfaces with small necksize to arbitrary points of any nondegenerate CMC surface. The proof uses the method of Cauchy data matching from \citeMP, cf. also \citeMPP. In the second part of this paper, we develop the consequences of this result and (at least partially) characterize the image of the map which associates to each complete, Alexandrov-embedded CMC surface with finite topology its associated conformal structure, which is a compact Riemann surface with a finite number of punctures. In particular, we show that this `forgetful' map is surjective when the genus is zero. This proves in particular that the CMC moduli space has a complicated topological structure. These latter results are closely related to recent work of Kusner \citeKu.