Constant mean curvature surfaces with Delaunay ends

Construction of compact constant mean curvature hypersurfaces with topology

“…In the case of complete noncompact constant mean curvature surfaces, the moduli space of such surfaces is now fairly well understood (in the genus 0 case). Then, many examples of such surfaces are produced in [9] and [16] and a classification of embedded constant mean curvature surfaces with three ends is given in [3]. However, the set of compact constant mean curvature is not so well understood.…”

“…The analogue of the following result for n = 2 is usually know as the "Linear Decomposition Lemma" (see [17] and [16]). …”

Generalized connected sum construction for scalar flat metrics

“…They have been used to understand solutions to problems arising from the geometry (minimal and constant mean curvature surfaces [13,14], constant scalar curvature metrics [9,12,15], and recently even Einstein metrics [1]) and from the physic (Einstein constraint equations [7] and [8]). However most of the results are concerned with the connected sum at points (point-wise connected sum), whereas the case of connected sum along a submanifold (generalized connected sum or fiber sum) has received less attention.…”

The dynamics theorem for CMC surfaces in $R^3$

“…The collection of properly embedded CM C surfaces with bounded second fundamental form is quite large and varied (see [4,10,11,15,16,17]). Many of these examples appear as doubly and singly-periodic surfaces.…”

“…Many of these examples appear as doubly and singly-periodic surfaces. The techniques of Kapouleas [10] and MazzeoPacard [16] can be applied to obtain many nonperiodic examples of finite and infinite topology. Some theoretical aspects of the study of these special surfaces have been developed previously in works of Meeks [18], Korevaar-Kusner-Solomon [13] and Korevaar-Kusner [12]; results from all of these three key papers are applied here.…”