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.


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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