Constant Mean Curvature Surfaces of Any Positive Genus

Abstract

Abstract. We show the existence of several new families of non-compact constant mean curvature surfaces: (i) singly-punctured surfaces of arbitrary genus g ≥ 1, (ii) doubly-punctured tori, and (iii) doubly periodic surfaces with Delaunay ends.

Holomorphic representation of constant mean curvature surfaces in Minkowski space: Consequences of non-compactness in loop group methods

“…Concerning CMC surfaces, notable early results were the classification of CMC tori in R 3 by Pinkall and Sterling [25], and the rendering of all CMC tori in space forms in terms of theta functions by Bobenko [5]. The DPW method has led to new examples of non-simply-connected CMC surfaces in R 3 -and other space forms -that have not yet been proven to exist by any other approach [22], [23], [29].…”

Constant mean curvature cylinders with irregular ends

“…This article takes the second approach, and presents a very general class of potentials which naturally encode the closing conditions, and for which there exists a very simple unitarizer. The general form of our potentials is the same as those of k-noids [11,12] with Delaunay ends, and higher genus surface with ends [6], and thus fits nicely into a general framework of potentials for non-compact cmc surfaces.…”

“…The inequality a 2 0 − a −1 a 1 > 0 guarantees that the trace along S 1 is decreasing from 2 at λ = 1. By a technique in [6], a rescaling of the potential…”

“…It follows that tr M c 0 ∈ [−2, 2] for λ ∈ S 1 , and is therefore unitarizable there. For similar uses of this technique see [6,2].…”

Construction of constant mean curvature n-noids using the DPW method

“…Assume that t = 0 and let x = (a i , b i , p i ) 1≤i≤n be a solution of Problem (11). From Equations (14) and (15) we infer that F n (0, x) = G n (0, x) = 0. Let i ∈ [1, n].…”