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**An end-to-end construction for compact constant mean curvature surfaces.**
*(English)*
Zbl 1110.53043

The theory of constant mean curvature surfaces in Euclidean space has been the object of intensive study in the past years. In this paper, the authors provide a construction for compact surfaces with constant mean curvature of genus 3 and higher, based on tools developed for the understanding of complete noncompact constant mean curvature surfaces and the end-to-end construction developed by Ratzkin (cf. [J. Ratzkin, “An end-to-end gluing construction for surfaces of constant mean curvature.” Ph.D. thesis, University of Washington, Seattle, (2001)] to connect (and produce) complete noncompact constant mean curvature surfaces along their ends. In contrast to the method of N. Kapouleas [Ann. Math. (2) 131, No. 2, 239–330 (1990; Zbl 0699.53007), Invent. Math. 119, No. 3, 443–518 (1995; Zbl 0840.53005)] this construction is technically simple. This parallels the fact that the end-to-end construction of J. Katzkin is simpler than earlier constructions of complete noncompact surfaces of constant mean curvature.

Reviewer: Shen Yi-Bing (Hangzhou)

### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |