Embedded constant mean curvature surfaces in Euclidean three-space

Abstract

Abstract. In this paper we refine the construction and related estimates for complete Constant Mean Curvature surfaces in Euclidean three-space developed in [10] by adopting the more precise and powerful version of the methodology which was developed in [14]. As a consequence we remove the severe restrictions in establishing embeddedness for complete Constant Mean Curvature surfaces in [10] and we produce a very large class of new embedded examples of finite topology.

On the non-existence of compact surfaces of genus one with prescribed, almost constant mean curvature, close to the singular limit

“…Let us point out that in the setting of CMC surfaces there are many remarkable results about the existence of complete, compact/non-compact, embedded/non-embedded surfaces with prescribed genus and a given number of ends (in the case of unbounded surfaces). Such surfaces are constructed by gluing segments or halves of Delaunay surfaces, according to techniques introduced by Kapouleas ( [20,21]) and subsequently adjusted or revised by other authors (e.g., [3,17,18,23,24]).…”

Solutions without any symmetry for semilinear elliptic problems

“…As explained above, we will not directly make use of this result in this paper. However, this result can, for example, be used to generalize the examples of solutions of (1.1) which infinite energy constructed by Malchiodi in [15] or it can also be used to construct complete non compact constant mean curvature surfaces in the spirit of [12,11]. 4.3.…”

“…Our main theorem is very much inspired from the construction of compact and complete, non compact constant mean curvature surfaces by Kapouleas [12,10,11]. Indeed, the construction of networks Z satisfying both (2.8) and (2.9) which we will describe in the next sections can be easily adapted to shed light on the configurations used by Kapouleas to construct both compact and non compact constant mean curvature surfaces and in fact this provides a systematic construction of flexible graphs used in [12,11] or c-graphs used in sections 2 and 3 of [10]. More precisely, what we call unbalanced flexible graphs are graphs which can be used to construct complete, non compact constant mean curvature surfaces and they can also be used to generalize the construction of infinite energy solutions of (1.1) by Malchiodi [15].…”

“…As we will see, in our case and in contrast with the analysis of [12,10,11], we need to restrict our attention to what we call embedded networks and we also have to handle some delicate issue which will be described in section 5. These are two additional constraints which are not present in the construction of compact (and complete, non compact) constant mean curvature surfaces.…”

The method of moving planes: a quantitative approach

“…We notice that Theorem 1.7 is not optimal from a qualitative point of view (see [28] and the reference therein and [31,32]), since the touching ball condition prevents the possibility of having a bubbling phenomenon. In this direction, Theorem 1.7 was improved in [39] where the bubbling phenomenon was characterized (see also [58] for the anisotropic counterpart).…”