The embedded Calabi-Yau conjecture for finite genus

The embedded Calabi-Yau conjecture for finite genus William H. Meeks III∗Joaqu´ ın P´ erez Antonio Ros† March 19, 2019 Abstract Suppose Mis a complete, embedded minimal surface in R3with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of Mhave properly embedded representatives with compact boundary, genus zero and with constrained geometry. We use this result to show that if Mhas at least two simple limit ends, then Mhas exactly two simple limit ends. Furthermore, we demonstrate that Mis properly embedded in R3if and only if Mhas at most two limit ends if and only if Mhas a countable number of limit ends. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42 Key words and phrases: Proper minimal surface, embedded Calabi-Yau problem, minimal lamination, limit end, injectivity radius function, locally simply connected. 1 Introduction. The Calabi-Yau conjectures refer to a series of conjectures concerning the nonexistence of a complete, minimally immersed surface f:M→R3whose image f(M)is constrained to be contained in a particular region of R3(see Calabi [2], page 212 in Chern [3], problem 91 in Yau [45] and page 360 in Yau [46]). Calabi’s original conjectures [2] state that a complete, nonflat minimal surface cannot be contained in the unit ball B(1) = {x∈R3| |x|<1} or even in a halfspace of R3. Among the positive results on the Calabi-Yau conjectures, we mention that the Strong Halfspace Theorem [17] implies the validity of the conjectures for properly immersed minimal surfaces in a closed halfspace. A spectacular positive result by Colding and Minicozzi [9] is that any complete, embedded minimal surface Min R3with finite topology is proper, and so the Halfspace Theorem (Hoffman and Meeks [17]) implies that Mcannot be contained in a halfspace unless it is a finite number of parallel planes. In contrast to Colding and Minicozzi’s properness result for the finite topology embedded CalabiYau problem, Ferrer, Mart´ ın, Meeks and Nadirashvili have conjectured that there is a particular bounded domain Ωin R3(see [14] for a description of Ω), which is smooth except at one ∗This material is based upon work for the NSF under Award No. DMS - 1309236. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the NSF. †Research partially supported by MINECO/FEDER grants no. MTM2014-52368-P and MTM2017-89677-P. 1 arXiv:1806.03104v1 [math.DG] 8 Jun 2018 point and satisfies the following property: Every open surface with compact (possibly empty) boundary whose ends have infinite genus admits a complete, proper minimal embedding into Ω. We refer the reader to Section 2for a brief elementary topological discussion of the notions of end, the genus of an end, limit end, simple limit end and end representative for any noncompact surface, terms we will use freely in this manuscript. The theory developed in this paper represents the first step in resolving the following fundamental conjecture, which gives a strong converse to the just mentioned existence conjecture of Ferrer, Mart´ ın, Meeks and Nadirashvili for open surfaces with compact boundary. Conjecture 1.1 (Embedded Calabi-Yau Conjecture for Finite Genus) Every connected, complete embedded minimal surface M⊂R3of finite genus and compact (possibly empty) boundary is properly embedded in R3. Corollary 1 in [30] implies Conjecture 1.1 under the additional hypothesis that Mis a leaf of a minimal lamination Lof R3, or equivalently, when Mhas locally bounded Gaussian curvature in R3. As mentioned above, Colding and Minicozzi [9] have proved Conjecture 1.1 under the additional assumption that Mhas finite topology. In [36], Meeks and Rosenberg proved that connected, complete embedded minimal surfaces in R3with positive injectivity radius are proper; their theorem is a generalization of the properness result of Colding and Minicozzi since complete, embedded finite topology minimal surfaces in R3have positive injectivity radius. These results, together with others by Bernstein and Breiner [1], Collin [11], Meeks and P´ erez [23] and Meeks and Rosenberg [35], imply that a complete, nonflat embedded minimal surface M⊂R3with finite topology has annular ends which are asymptotic to ends of planes, catenoids or Mhas just one end which is asymptotic to the end of a helicoid. In all of these cases, Mis proven to be conformally a compact Riemann surface Mpunctured in a finite number of points (in particular, Mis recurrent for Brownian motion), and the embedding of M into R3can be expressed analytically in terms of meromorphic data defined on M. In the case that a complete embedded minimal surface Mof finite topology in R3has nonempty compact boundary, a similar description of its conformal structure (∂M has full harmonic measure) and of its asymptotic behavior (a few more asymptotic types arise than in the case without boundary) hold, see [23] for details. Concerning conformal questions, one consequence of the results in this paper is Corollary 1.8, which states that if a properly embedded minimal surface in R3has a limit end of genus zero, then it is recurrent; this can be viewed as a generalization of our previous result [31] that any properly embedded minimal surface of finite genus in R3is recurrent. Using the techniques developed by Colding and Minicozzi [5,6,7,8,9,10], Meeks and Rosenberg [36] and those in our papers [26,28,30,31,34], we shall prove here that if a complete, embedded minimal surface of finite genus in R3has a countable number of limit ends, then it is properly embedded in R3(see Theorem 1.3 below). By the main result of Collin, Kusner, Meeks and Rosenberg in [12], any properly embedded minimal surface in R3 must have a countable number of ends, even if it does not have finite genus. More generally, 2 the results in [12] imply that a properly embedded minimal surface with compact boundary in R3can have at most two limit ends, and that if it has empty boundary and two limit ends, then it is recurrent. Our first key partial result on Conjecture 1.1 is the following theorem, which is the main result in Section 3(see Remark 3.4). Theorem 1.2 Let M⊂R3be a complete embedded minimal surface of finite genus with compact boundary and exactly one limit end. Then Mis properly embedded in R3. More generally, we have the following extension of the above result, which is proved in Section 5. Theorem 1.3 Suppose M⊂R3is a complete, connected, embedded minimal surface of finite genus, an infinite number of ends and compact boundary (possibly empty). Then: 1. Mhas at most two simple limit ends. 2. Mhas exactly one or two limit ends if and only if Mis proper in R3. 3. Suppose Mhas a countable number of limit ends. Then: 3-A. Mhas one or two limit ends. 3-B. Mis proper in R3. 3-C. If Mhas two limit ends, then its annular ends are planar. 3-D. If ∂M =Ø, then Mhas exactly two limit ends and Mis recurrent for Brownian motion. 3-E. If ∂M 6=Ø, then ∂M has full harmonic measure. Remark 1.4 In contrast to item 3-D of Theorem 1.3, we note that Traizet [43] has constructed a complete embedded minimal surface in R3of infinite genus, with empty boundary, one limit end and infinitely many catenoidal type ends. The proof of Theorem 1.3 depends on Theorem 1.6 below, which describes the geometry, topology and conformal structure of certain representatives of a simple limit end of genus zero for a complete embedded minimal surface in R3; see Figure 1for a suggestive picture describing the key geometric features of such a representative. Before stating Theorem 1.6, we will need the following definition. Definition 1.5 Let Ebe a complete embedded minimal surface in R3with nonempty compact boundary ∂E. We define the flux vector of Eas FE=Z∂E η∈R3,(1) where ηdenotes the inward pointing unit conormal vector to Ealong ∂E. 3 DE (x1, x2)−plane genus zero FE= (h, 0,1) Flux along boundary 1 2 3 4 5 simple limit end simple ends logarithmic growths ≤0 Figure 1: A graphical representation of the end representative Ein Theorem 1.6. Theorem 1.6 Suppose eis a simple limit end of genus zero of a complete, connected, embedded minimal surface M⊂R3with compact (possibly empty) boundary. Then ecan be represented by a subdomain E⊂Int(M)that is properly embedded in R3and, after a translation, rotation and homothety of M,Esatisfies the following statements: 1. The annular ends of Ehave nonpositive logarithmic growths. 2. Ehas genus zero and one limit end, which, in the natural ordering of the ends of Egiven by the Ordering Theorem1in [16], is the top end of E. 3. The boundary ∂E is a simple closed curve in the (x1, x2)-plane, and the flux vector FEof Edefined as in (1) is (h, 0,1) for some h > 0. Furthermore, ∂E bounds a convex disk DE⊂ {x3= 0}whose interior is disjoint from E, see Figure 1. 4. There exists an orientation preserving diffeomorphism f:R3→R3such that f(R+) = E, where R+is the top half of a Riemann minimal example2with boundary circle in the (x1, x2)-plane. 5. Ehas bounded Gaussian curvature. 6. Eis conformally diffeomorphic to the closed punctured disk {z∈C|0<|z| ≤ 1}minus a sequence of points converging to 0. In particular, ∂E has full harmonic measure. 1Observe that the Ordering Theorem stated in [16] also holds for properly embedded minimal surfaces in R3 with compact boundary. 2See [29,33] for a discussion of the singly-periodic, genus-zero, Riemann minimal examples. 4 Remark 1.7 If M⊂R3is a properly embedded minimal surface of finite genus and infinite topology, then Mhas exactly two limit ends e−∞, e∞which are simple limit ends of genus zero, and which admit representatives E−∞, E∞that satisfy the conclusions of Theorem 1.6, see [30,31]. In this case where Mhas no boundary, then Theorem 8.1 in [33] implies that the asymptotic behavior of each of its two limit ends can be described by the geometry of the ends of a Riemann minimal example. Crucial ingredients in the proof of Theorem 1.6 are the Limit Lamination Closure Theorem (Theorem 1 in [36]), the Local Picture Theorem on the Scale of Topology (Theorem 1.1 in [27]) and Theorem 2.2 in [26] on the structure of certain possibly singular lamination limits of certain sequences of minimal surfaces in R3. These ingredients, as well as many arguments in this paper, rely heavily on Colding-Minicozzi theory. Theorems 1.3 and 1.6 not only play an important theoretical role in our strategy to prove Conjecture 1.1, but they also have important consequences for properly embedded minimal surfaces, such as the one given in the next corollary; this corollary follows from the more general result Corollary 6.1. Corollary 1.8 If M⊂R3is a properly embedded minimal surface with a limit end of genus zero, then Mis recurrent for Brownian motion. Some of the results in this paper were announced by the authors at a conference in Paris in 2004. Our proofs use results of Colding-Minicozzi theory that led us to develop a detailed study of minimal laminations with singularities and subsequent applications. The present paper can be considered as a culmination of a long term project in the understanding of complete embedded minimal surfaces of finite genus in R3. 2 Preliminaries on the ends of a noncompact surface. Given p∈R3and R > 0, we denote by B(p, R)the open ball centered at pof radius R. When p=~ 0, we let B(R) = B(~ 0, R). If Σ⊂R3is a surface and p∈Σ, then KΣ, dΣ, IΣ, BΣ(p, R) and TpΣrespectively stand for the Gaussian curvature function of Σ, its intrinsic distance function, its injectivity radius function, the intrinsic ball centered at pof radius R > 0and the tangent plane to Σat p. Also, D={z∈C:|z| ≤ 1}stands for the closed unit disk. Throughout the paper, M⊂R3will denote a connected, complete embedded minimal surface with compact boundary (possibly empty). We next recall the notion of end of M. Consider the set A={α: [0,∞)→M|αis a proper arc}. In A, we define the equivalence relation α1∼α2if for every compact set C⊂M,α1, α2lie eventually (outside a compact subset of the parameter domain [0,∞)) in the same component of M−C. 5 Definition 2.1 Each equivalence class in E(M) = A/∼is called an end of M. If e∈ E(M), α∈eis a proper arc and E⊂Mis a proper connected subdomain with compact boundary such that α([t0,∞)) ⊂Efor some t0≥0, then we say that Erepresents the end e. The space E(M)has the following natural Hausdorff topology. For each proper subdomain E⊂Mwith compact boundary, we define the basis open set B(E)⊂ E(M)to be those equivalence classes in E(M)which have representative proper arcs contained in E. With this topology, E(M)is a totally disconnected compact space which embeds topologically as a subspace of [0,1] ⊂R(see pages 288-289 of [24] for a proof of this property). In the sequel, we will view E(M)as a subset of [0,1] endowed with the induced metric topology. Note that every simple end xof M(i.e., xis an isolated point of E(M)) with genus zero can be represented by a proper annulus Ex⊂Mwhich is homeomorphic to S1×[0,∞). Next consider a simple limit end eof M, i.e., there exists a neighborhood O(e)⊂ E(M) such that O(e)− {e}consists of simple ends and eis a limit point of a sequence of simple ends {xn}n⊂O(e)−{e}. Suppose that the simple limit end ehas genus zero, i.e., eadmits a representative of genus zero. By the classification of genus-zero surfaces and after a possible replacement by a smaller neighborhood O(e)of ein E(M), there exists a proper subdomain Eof Msatisfying: (A1) Eis diffeomorphic to D(∗) = D−[0}∪{1 2nn∈N]. Furthermore, ∂E ∩∂M =Ø. (A2) Erepresents all the ends in O(e), and the equivalence class under ∼of every proper arc in Erepresents a unique end in O(e). 3 Simple limit ends of genus zero can be represented by properly embedded surfaces. We begin this section with several key notions that are closely tied to obtaining properness results for minimal surfaces, including Theorem 1.2 which will be proved here. Definition 3.1 1. An embedded surface with boundary (possibly empty) Σ⊂R3is said to be locally simply connected in R3if for every p∈R3, there exists r=r(p)>0such that the closure of each component of Σ∩B(p, r)that is disjoint from ∂Σ, is a compact disk with boundary in ∂B(p, r).Σhas locally positive injectivity radius away from ∂Σif for every p∈R3, there exists r=r(p)>0such that the injectivity radius function IΣof Σ is bounded away from zero on the union of the components of Σ∩B(p, r)that are disjoint from ∂Σ. 2. Let A⊂R3be an open set and {Σn}n∈N⊂R3be a sequence of embedded surfaces (possibly with boundary). The sequence {Σn}nis called locally simply connected in Aif for every p∈A, there exists r=r(p)>0such that B(p, r)⊂Aand for nsufficiently large, B(p, r)intersects Σnin components that are disks with boundaries in ∂B(p, r).{Σn}nis said to have locally positive injectivity radius in A,if for every p∈A, there exists εp>0 6 and np∈Nsuch that for n > np, the restricted functions (IΣn)|Σn∩B(p,εp)are uniformly bounded away from zero. Remark 3.2 With the notation of item 2 of Definition 3.1, if the surfaces Σnhave nonempty boundaries and {Σn}nhas locally positive injectivity radius in A, then for any p∈Athere exists εp>0and np∈Nsuch that ∂Σn∩B(p, εp) = Ø for n>np, i.e., points in the boundary of Σnmust eventually diverge in space or converge to a subset of R3−A. By Proposition 1.1 in [9], if M⊂R3is an embedded minimal surface, then the property that Mis locally simply connected in R3is equivalent to the property that Mhas locally positive injectivity radius away from ∂M. The same proposition gives that a sequence of embedded minimal surfaces {Mn}nhas locally positive injectivity radius in an open set A⊂ R3if and only if {Mn}nis locally simply connected in A. Theorem 2 in [36] implies that if an embedded, complete, nonflat minimal surface in R3 (with empty boundary) has positive injectivity radius, then it is proper. Although not stated explicitly in [36], the following result is an immediate consequence of the proof of Theorem 2 in [36] and other arguments therein. Theorem 3.3 Let M⊂R3be a complete, connected, embedded minimal surface with compact boundary. If the injectivity radius function IMof Mis bounded away from zero outside of some intrinsic ε-neighborhood of ∂M, then Mis proper in R3. Furthermore, if Mhas finite topology, then IMis bounded away from zero outside of some intrinsic ε-neighborhood of ∂M, and so, Mis proper in R3. Remark 3.4 Theorem 3.5 below implies the main properness statement in Theorem 1.6. It also implies Theorem 1.2 by the following reasoning. Suppose that M⊂R3is a complete embedded minimal surface of finite genus with compact boundary and exactly one limit end e, which must therefore be a simple limit end. If Eis the proper representative of egiven in the next theorem, then the surface M−Int(E)has finite topology and must therefore be proper by Theorem 3.3; hence, M=E∪(M−Int(E)) is also proper in R3. Theorem 3.5 Suppose that M⊂R3is a complete embedded minimal surface with possibly empty compact boundary. Every simple limit end e∈ E(M)of genus zero can be represented by a subdomain E⊂Mwith compact boundary whose injectivity radius is bounded away from zero outside each compact neighborhood of its boundary. In particular, Eis proper in R3. Proof. Let e∈ E(M)be a simple limit end of genus zero. Consider a proper subdomain E⊂Msatisfying properties (A1) and (A2) stated in the preliminaries section. With a slight abuse of notation, we identify Ewith the parameter domain D(∗) = D−[0}∪{1 2nn∈N], see property (A1). The proof of Theorem 3.5 will be divided into several statements; more precisely, Lemmas 3.7 and 3.11 and Propositions 3.8 and 3.12. As the proof develops, we will replace Eby similar proper subdomains of Eand O(e)by the open subset of ends of the replaced E, but will continue to label these objects by the same letters. 7 We first deal with the (simple) annular ends in E. For n∈N, let S1 ndenote the circle of center 0∈Cand radius 1 2n+1. Let Enbe the proper subdomain of Ebounded by ∂E ∪S1 n. Since Enhas finite topology and compact boundary, then Theorem 3.3 applied to Eninsures that Enis proper in R3. As each of the (finitely many) ends of Enis an annular end, then Collin’s theorem [11] implies that each end of Enhas finite total curvature and is asymptotic to an end of a plane or catenoid. After a rigid motion in R3, we may assume that: (B1) The annular ends of Eare represented by graphs over their projections to {x3= 0}with logarithmic growth (which is zero when the end is asymptotic to the end of a plane). The proof of Theorem 3.5 is by contradiction. Hence assume there is no end representative Eof ewhich is a proper surface. By Theorem 3.3, the injectivity radius function IEof every such a representative has the property that IEfails to be bounded away from zero outside some small ε-neighborhood of ∂E. Therefore, there exists a sequence of points qn∈Esuch that dE(qn, ∂E)is bounded away from zero and IE(qn)→0as n→ ∞. Clearly, the qn diverge in E. As IEbecomes unbounded when approaching each of the simple ends of E, we deduce that the qnconverge to the origin when viewed inside D(∗). Since Ehas genus zero, then the Local Picture Theorem on the Scale of Topology (see Theorem 1.1, Proposition 4.20 and Remark 4.32 in [27]) implies that we can find a divergent sequence of points pn∈E(called points of almost minimal injectivity radius for E) and positive numbers εn→0, such that dE(pn, qn)→0as n→ ∞and: (C1) The closure Mnof the component of B(pn, εn)∩Ethat contains pnis compact with boundary ∂Mn⊂∂B(pn, εn). Furthermore, Mnis disjoint from ∂E for nlarge enough (this follows from the fact that pnis divergent in E). (C2) Let λn= 1/IMn(pn), where IMndenotes the injectivity radius function of Erestricted to Mn. Then, λnIMn≥1−1 nin Mnand λnεn→ ∞. Furthermore, exactly one of the following two cases occurs after extracting a subsequence. (C3) The surfaces λn(Mn−pn)have uniformly bounded Gaussian curvature on compact subsets of R3. In this case, there exists a connected, properly embedded minimal surface M∞⊂R3with ~ 0∈M∞,IM∞≥1and IM∞(~ 0) = 1, such that for any k∈N, the surfaces λn(Mn−pn)converge Ckon compact subsets of R3to M∞with multiplicity one as n→ ∞. (C4) After possibly a rotation in R3, the surfaces λn(Mn−pn)converge to a minimal parking garage structure3of R3consisting of a foliation Fof R3by horizontal planes, with two columns l1, l2such that the associated highly sheeted, double multivalued graphs forming in λn(Mn−pn)around l1, l2for nsufficiently large, are oppositely handed. Furthermore, after relabeling, l1intersects B(1) and l2is at distance 1 from l1. 3We refer the reader to Section 3 in [27] for the definition of parking garage structure of R3. 8 In order to finish the proof of Theorem 3.5, we must find a contradiction in each of the Cases (C3), (C4) above. Suppose first that Case (C3) holds. As the λn(Mn−pn)all have genus zero, then M∞has genus zero as well. By classification results for properly embedded minimal surfaces of genus zero (Collin [11], L´ opez-Ros [18], Meeks-P´ erez-Ros [33]), M∞is a catenoid or a Riemann minimal example. Let γ⊂M∞be the waist circle if M∞is a catenoid, and in the case M∞ is a Riemann minimal example, then let γbe a simple closed planar curve (actually a circle) which separates the two limit ends of M∞. Remark 3.6 In the sequel, we will need the notion of flux vector of a minimal surface along a closed curve Γonce we have chosen a unit conormal vector ηalong Γ; this flux is the vector in R3given by the integral of ηalong Γ, which clearly is defined up to a sign. This ambiguity still lets us make sense of when this flux is nonzero, or when it is vertical. Since γhas nonzero flux, then for nlarge, γis approximated by the image by the composition of a translation by vector −pnwith a homothety by λnof a simple closed planar curve γn⊂Mnalso with nonzero flux. Lemma 3.7 γ⊂M∞has vertical flux. Furthermore, after choosing a subsequence, each curve γnalso has vertical flux. Proof. It suffices to prove that for nlarge, γn⊂Ehas vertical flux. If the subdisk in D bounded by γndoes not contain 0∈D, then γnis homologous to a finite number of loops around the annular ends of E, and so, γnhas vertical flux by property (B1). Otherwise, after replacing by a subsequence, we may assume that γnis topologically parallel to γn+kand to ∂E in D− {0}for n, k ∈N,nlarge. Hence for any k∈N,γnis homologous in Eto the union of γn+kwith a finite number of loops around annular ends of E, and so, the flux along γnis equal to a vertical vector minus the flux along γn+k. Since the flux along γn+kgoes to zero as k→ ∞(because length(γn+k)→0), then the flux of Ealong γnis vertical. 2 Proposition 3.8 M∞is not a Riemann minimal example. Proof. Arguing by contradiction, assume M∞is a Riemann minimal example. Since γ⊂M∞ has nonzero vertical flux by Lemma 3.7, then Theorem 6 in [30] implies that the planar ends of M∞are not horizontal. Let Qbe the plane passing through the origin in R3that is parallel to the planar ends of M∞ (equivalently, Qis the limit tangent plane at infinity of M∞). Observe that planes parallel to Qat heights (with respect to Q) different from the heights corresponding to the planar ends of M∞, intersect M∞transversely in simple closed curves (actually in circles). Let Γ1,Γ2,Γ3,Γ4 be four such circles on M∞, chosen so that the cycles Γ1∪Γ2,Γ2∪Γ3and Γ3∪Γ4each bound a noncompact subdomain Ω1,2,Ω2,3,Ω3,4respectively of M∞, each containing exactly two planar ends and such that Ω1,2∩Ω2,3= Γ2and Ω2,3∩Ω3,4= Γ3, see Figure 2top. Observe 9 Proof. Since we are in Case (D3), then γnbounds a proper annulus R(n)⊂E. After replacing γnby one of the boundary curves of the almost perfectly formed catenoid Cn, we have that the new annulus R(n)⊂Ewith ∂R(n)the replaced boundary curve, is disjoint from Int(Cn), and thus, we can assume that the total absolute curvature of R(n)is arbitrarily small for n sufficiently large. Since the Gauss map of R(n)is open, almost vertical along ∂R(n)(by Lemma 3.7), the image of this Gauss map has a limiting value (0,0,±1) at the end of R(n), and the spherical image of the Gauss map of R(n)is arbitrarily small, then we deduce that R(n)is the graph of a function defined on the projection of R(n)to the (x1, x2)-plane, and this graph has arbitrarily small gradient. As we can assume that γn→p∞as n→ ∞, it follows that the graphical annuli R(n)converge smoothly away from p∞to the horizontal plane L(p∞)passing through p∞. To finish the proof of the assertion, it only remains to show that Int(E)∩L(p∞) = Ø. Arguing by contradiction, suppose that L(p∞)intersects Eat an interior point. Since L(p∞)is not contained in E, then L(p∞)intersects Etransversely at some interior point of E. This implies that for n sufficiently large, R(n)intersects E−R(n), which is impossible since Eis embedded. Now the assertion is proved. 2 We next check that Case (D2) does not occur for nlarge. Arguing by contradiction, assume that nis large and (D2) holds. Notice that for nfixed and for k≥1, the proper subdomains R(n, k)bounded by γn∪γn+kgive rise to an proper exhaustion of the representative of the limit end of Ewhose boundary is γn. Rather than choosing γnnear the waist circle of the forming unstable compact catenoid piece Cn, we choose γnto be a curve contained in a horizontal plane at a height so that for each k, the (noncompact) subdomain R(n, k)⊂Econtains two unstable, pairwise disjoint, compact almost-catenoidal pieces, also denoted by Cn,Cn+k, near γnand γn+krespectively, so that Cnis an annular neighborhood of γn(resp. Cn+kis a neighborhood of γn+k) in the new proper domain R(n, k). We may assume that both boundary curves of Cnand of Cn+kare convex horizontal curves for all k. Also, ncan be chosen so that for all ksufficiently large, the almost-catenoid Cn+kis much smaller than the scale of the almost-catenoid Cn, see Figure 5. Given n∈N, let D0 n⊂R3be the horizontal open disk bounded by ∂Cn−γn(recall that Dnis the horizontal open disk bounded by γn). We next analyze the intersection of R(n, k) with Dn, D0 n, Dn+k, D0 n+k. (D2-a) We may assume that Dn+k, D0 n+kare disjoint from Cn(because the scale of Cn+kis much smaller than the scale of Cn, and both Cn, Cn+kare inside Ewhich is an embedded surface). (D2-b) An analogous reasoning as in the proof of Assertion 3.13 shows that both Dn+k,D0 n+k are disjoint from R(n, k). Observe that the boundary curve γn+kmust be contained in the compact region Wn⊂R3bounded by Cn∪Dn∪D0 nas in Figure 5 (otherwise the arguments in the proof of Assertion 3.13 lead to a contradiction). The maximum principle and the fact that the scale of Cn+kis much smaller than the scale of 16 CnCn+k Dn+k γn+k Dn γn D0 n+k D0 n Figure 5: The boundary curve γn+kof R(n, k)must be contained in the compact region Wn⊂ R3bounded by Cn∪Dn∪D0 n. 17 Cnimply that Cn+kis contained in the interior of Wn. Therefore, the topological balls Wncan be assumed to be concentric, in the following sense: (?)After replacing by a subsequence and re-indexing, Wn+1 ⊂Int(Wn). Since the scales of the catenoids Cnare converging to zero as n→ ∞, Property (?)implies that the Wnconverge to a point c∞∈R3, which satisfies {c∞}=Tn∈NWn⊂Int(W1). Without loss of generality, we may assume that ∂E ∩W1=Ø. We next prove that the surface E(W1) := E∩[Int(W1)− {c∞}]has locally positive injectivity radius in Int(W1)− {c∞}. Otherwise, there is a point q∈Int(W1)− {c∞}and a sequence of points qj∈E(W1),j∈N, of almost minimal injectivity radius for E(W1) in the sense of the Local Picture Theorem on the Scale of Topology, that diverge in E(W1) but converge to qas j→ ∞. After blowing up E(W1)around the points qjon the scale of the injectivity radius, we find a limit which is a catenoid (i.e., the other possibilities given by the Local Picture Theorem on the Scale of Topology are not possible by the arguments in Proposition 3.8 and Lemma 3.11). In particular, the catenoid which is forming nearby qjinside E(W1)for jlarge, is of one of the types (D1), (D2) or (D3); in this case we will simply say that Case (D1), (D2) or (D3) holds for qj. Case (D1) for qjis not possible by our previous arguments based on Assertion 3.13 and the L´ opez-Ros deformation. Also observe that Case (D2) cannot occur at qjfor jlarge, because the qjare converging to q6=c∞, which implies that qjdoes not lie in Wnfor nlarge but fixed, in contradiction with Property (?). This implies that for jlarge, Case (D3) holds for qj. Since the qjconverge to qand Case (D3) holds for qjfor every j, then Assertion 3.14 insures that the horizontal plane L(q)passing through qin disjoint from Eafter removing any small compact neighborhood of ∂E. This is impossible, since L(q) intersects C1. This contradiction proves that E(W1)has locally positive injectivity radius in Int(W1)−{c∞}. Since E(W1)has locally positive injectivity radius in Int(W1)−{c∞}, Remark 2 in [36] ensures that the closure of E(W1)in Int(W1)−{c∞}is a minimal lamination Lof Int(W1)− {c∞}that contains E(W1)as a subcollection of leaves. We next prove that Lhas no limit leaves in some neighborhood of c∞. Otherwise, the sublamination L0of limit leaves of Lis not empty, and L0consists of stable leaves by Theorem 1 in [32]. By Corollary 7.1 in [34], L0extends across c∞to a lamination of Int(W1). Thus, there exists a stable minimal surface L1⊂Int(W1)passing through c∞such that L1− {c∞}is a leaf of L0. Since L1is stable and Cnis unstable, then L1is disjoint from Cnfor all n≥2. Therefore, for ε > 0small enough, the ball B(c∞, ε)of center c∞and radius εintersects L1 in a component Ω1which is a disk that separates B(c∞, ε). Take n∈Nlarge enough so that Wn⊂B(c∞, ε), which exists since {c∞}=Tn∈NWn. As Ω1contains c∞∈Int(Wn)but Ω1∩Cn=Ø and Wn∩∂Ω1=Ø, then Ω1∩(Dn∪D0 n)is nonempty. Without loss of generality, we may assume that Ω1intersects Dn∪D0 ntransversely and so, there exists a simple closed curve βin Ω1∩(Dn∪D0 n). This contradicts the maximum principle applied to the subdisk of Ω1bounded by β. This contradiction proves that Lhas no limit leaves in some neighborhood of c∞. Since Lhas no limit leaves in some neighborhood of c∞, we may assume that in some 18 small compact neighborhood Nof c∞in R3,L∩N= [E−{c∞}]∩Nand [E−{c∞}]∩N is a properly embedded minimal surface in N− {c∞}of genus zero. But properly embedded minimal surfaces of finite genus in a punctured Riemannian ball extend smoothly across the puncture (see for example, Corollary 2.7 in [28] for this minimal lamination extension result). This is clearly not possible because the Gaussian curvature of Eis not bounded in any neighborhood of c∞.This contradiction proves that Case (D2) does not occur for nlarge. Finally we check that Case (D3) does not occur, which will finish the proof of Proposition 3.12. By Lemmas 3.7,3.11 and Proposition 3.8 and from the previously considered cases, we may assume that all local pictures Mnof Eon the scale of topology (defined by properties (C1)-. . . -(C4)) produce, after blowing-up, limiting catenoids with vertical axes, and the horizontal almost waist circles γn⊂Eare in Case (D3) for all n∈N(after passing to a subsequence). Consider the related sequences {Mn}n,{γn}n. We can assume that for all n,Mn contains a compact piece of an almost perfectly formed unstable catenoid Cncontaining γn, where Cnis a shrunken image of a large compact portion of an almost-catenoid whose boundary consists of simple closed convex horizontal planar curves. Since we are in Case (D3), then γnbounds a proper annulus R(n)⊂E. After replacing γnby one of the boundary curves of the almost perfectly formed catenoid Cn, we have that the new annulus R(n)⊂Ebounded by γnsatisfies the following properties (see the proof of Assertion 3.14): (E1) R(n)is the graph of a function defined on the projection of R(n)to the (x1, x2)-plane, and this graph has arbitrarily small gradient. (E2) Length(γn)→0as n→ ∞. We will next show that Assertion 3.13 holds in this new setting. Assertion 3.15 After extracting a subsequence and possibly replacing Eby another end representative, for every n∈N, the open planar disks D1(n), D2(n)⊂R3bounded by the curves in ∂Cn, are disjoint from E. Proof. Let Wn⊂R3be the compact region bounded by Cn∪D1(n)∪D2(n). After choosing a subsequence and removing a small neighborhood of ∂E from E, we may assume that Wn∩ ∂E =Ø. Observe that E∩Int(Wn)is locally simply connected: otherwise, there exists some point p∞∈E∩Int(Wn)where Case (D3) holds for γmfor all m∈Nsufficiently large (m larger than n); in this case, Assertion 3.14 ensures that the horizontal plane L(p∞)passing through p∞is disjoint from Eafter removing any compact neighborhood of ∂E, which is impossible since L(p∞)∩Cn6=Ø. Thus, E∩Int(Wn)is locally simply connected. The arguments in the previous paragraph and Assertion 3.14 ensure that there exists an open set U⊂R3such that Cn⊂Uand the restriction of the injectivity radius function of E to E∩Uis bounded away from zero. Therefore, the closure of E∩Urelative to the open set Uis a minimal lamination of U. As E∩Int(Wn)is locally simply connected, the closure of E∩Int(Wn)relative to Int(Wn)is a minimal lamination of Int(Wn). Consequently, the closure of E∩[U∪Int(Wn)] is a minimal lamination of U∪Int(Wn). Since Cnis unstable, then Cn 19 is not contained in a limit leaf of this lamination, which implies that the distance from Cnto the closure E∩Int(Wn)of E∩Int(Wn)is positive. As Cnis unstable, we can find a compact unstable subannulus C0 n⊂Int(Cn)such that ∂C0 n consists of two convex horizontal curves that bound open planar disks D0 1(n), D0 2(n)⊂R3. Let W0 n⊂Wnbe the compact region bounded by C0 n∪D0 1(n)∪D0 2(n). It follows from the previous paragraph that the closure of E∩Int(Wn)relative to Int(Wn)is a minimal lamination of Int(Wn), that is at a positive distance from Cn. In particular, the closure of E∩Int(Wn) relative to Int(Wn)intersected with W0 nis a compact, possibly empty, set Xin W0 n. Suppose the assertion fails for some n, that is, Eintersects D1(n)∪D2(n). Then, E∩ Int(Wn)6=Ø and thus, we can assume E∩Int(W0 n)6=Ø by choosing C0 nsufficiently close to Cn. In particular, X6=Ø. As Xis a compact union of minimal surfaces in W0 n, then the maximum principle applied to x3gives that each component of Xintersects both disks D0 1(n), D0 2(n). Since Xis a good barrier for solving Plateau type problems in W0 n, and ∂C0 n does not bound minimal disks in W0 n−X, then there exists a least area annulus A0⊂W0 n with boundary ∂A =∂C0 n. This is impossible, by the same reasoning as in the proof of Assertion 3.13. This completes the proof of Assertion 3.15.2 Arguing by contradiction, assume that Case (D3) occurs for all n. By our earlier considerations, there would exist an infinite collection of pairwise-disjoint almost-catenoids Cnforming on Eof the type described in Case (D3) and that satisfy the conclusions of Assertion 3.15. Also, we can assume that the logarithmic growths of the associated graphs R(n)all have the same sign, say negative. Consider the piecewise smooth graphical planes Pn=D2(n)∪R(n), where D2(n)is the lower open disk given in Assertion 3.15. Note that as D2(n)∩E=Ø, then E−R(n) is contained in the component of R3−Pnabove Pn. It follows that the connected surface E−∪nR(n)must lie above each of the Pn. By elementary separation properties, this situation is not possible as it would imply that P1lies above P2and P2lies above P1.This contradiction completes the proof that Case (D3) does not occur. So, Proposition 3.12 is proved. 2 By Lemma 3.11, Propositions 3.8,3.12 and the paragraph before Remark 3.6, we conclude that the injectivity radius function IEis bounded away from zero outside of some (and thus, every) intrinsic ε-neighborhood of ∂E. Therefore, Theorem 3.3 insures that Eis properly embedded in R3, which completes the proof of Theorem 3.5.2 4 The proof of Theorem 1.6. Let ebe a simple limit end of genus zero of a complete, embedded minimal surface M⊂R3 with compact boundary (possibly ∂M =Ø). By Theorem 3.5, we can choose a representative Eof esuch that Eis properly embedded in R3. The arguments at the end of Section 2show that after relabeling, properties (A1), (A2) hold for E. As explained in the second paragraph of the proof of Theorem 3.5, each simple end of Ehas and an annular end representative with finite total curvature and is asymptotic to an end of a plane or catenoid, which after a fixed 20 rotation of Min R3, is a graph over its projection to the (x1, x2)-plane. Since Eis properly embedded in R3, it follows from the Ordering Theorem [16] and Theorem 1.1 in [12] that the limit end of E, after a possible rotation by πaround the x1-axis, is the top end of E. Lemma 3.6 in [12] implies that a limit end of a properly embedded minimal surface with compact boundary in R3cannot have a representative that lies above the end of a catenoid with positive logarithmic growth. Therefore, since the limit end of Eis its top end and the middle ends of Eare asymptotic to planes and catenoidal ends, none of the catenoidal ends in Ehave positive logarithmic growth. This proves items 1 and 2 of Theorem 1.6. Lemma 4.1 There exists a divergent sequence of points qn∈Esuch that IE(qn) |qn|→0as n→ ∞, where IEis the injectivity radius function of E. Proof. Otherwise, there exists c > 0such that IE(·)≥c|·|in E, away from a compact neighborhood of ∂E. Since ∂E is compact, Eis properly embedded and Edoes not have finite total curvature, then Theorem 1.2 in [34] implies that there exists a divergent sequence of points yn∈Esuch that KE(yn)|yn|2→ −∞ as n→ ∞. Consider the sequence of positive numbers σn=1 |yn|→0. Since IσnE(σnx) |σnx|=IE(x) |x|, we conclude that the sequence of surfaces {σnE}nhas locally positive injectivity radius in the open set R3−{~ 0}in the sense of Definition 3.1, or equivalently, the sequence of compact genuszero minimal surfaces {(σnE)∩B(n)}nis locally simply connected in R3−{0}, see the first paragraph after Remark 3.2. Since the surfaces (σnE)∩B(n)have genus zero with compact boundary and the Gaussian curvature of (σnE)∩B(n)at the point σnyn∈∂B(1) diverges as n→ ∞, then item 2 of Theorem 2.2 in [26] implies that after passing to a subsequence, {(σnE)∩B(n)}nconverges to a minimal lamination Lof R3− {~ 0}, outside of a nonempty singular set of convergence S(L)⊂ L (this is the closed subset of points x∈ L such that the supremum of the absolute Gaussian curvature of (σnE)∩B(x, ε)is not bounded in n, for any ε > 0), and the following property holds: (F) The closure Lof Lin R3is a foliation of R3by planes, and the closure S(L)of S(L) consists of one or two complete lines orthogonal to the planes in L. Since the limit end of Eis its top end and its annular ends are catenoidal with nonpositive logarithmic growth, it follows that Lis contained in the closed upper halfspace {x3≥0}minus the origin. This contradicts property (F) above, which completes the proof of Lemma 4.1.2 Consider the divergent sequence {qn}n⊂Egiven by Lemma 4.1. We next apply a similar rescale-by-topology argument as as we did in the proof of Theorem 3.5 just after property (B1), but instead of using the Local Picture Theorem on the Scale of Topology as we did there, we will use the following extrinsic argument. Given n∈Nlarge so that the boundary of Elies in 21 B(|qn|/2), consider the continuous, nonnegative function hn:B(qn,|qn|/2) ∩E→Rgiven by hn(x) = distR3(x, ∂B(qn,|qn|/2)) IE(x). hnvanishes at ∂B(qn,|qn|/2). Let pnbe a maximum of hn. Observe that hn(pn)≥hn(qn) = |qn| 2IE(qn)→ ∞, and define rn=1 2distR3(pn, ∂B(qn,|qn|/2)) = 1 2hn(pn)IE(pn). Then, the sequence of embedded minimal surfaces of genus zero and compact boundary e En=λnE∩B(pn, rn)−pn(2) is uniformly locally simply connected in R3, where λn= 1/IE(pn)(in fact, e Enhas boundary in the sphere centered at the origin with radius 1 2hn(pn)→ ∞ and the injectivity radius function of e Enis at least 1/2at points at least at distance 1/2from its boundary). By Theorem 2.2 in [26] applied to this sequence of surfaces, we deduce that there exists a minimal lamination Lof R3and a closed subset S(L)⊂ L such that {e En}nconverges Cβ, for all β∈(0,1), on compact subsets of R3−S(L)to L; here S(L)is the singular set of convergence of the e Ento L. Furthermore, exactly one of the two following cases holds: (G1) The surfaces e Enhave uniformly bounded Gaussian curvature on compact subsets of R3. In this case, S(L) = Ø and either Lis a collection of planes (this case cannot occur since the injectivity radius function of e Enat the origin is 1 for each n∈N), or Lconsists of a single leaf M∞, which is properly embedded in R3with genus zero. Furthermore, in this last case e Enconverges smoothly on compact sets in R3to M∞with multiplicity one and exactly one of the following three cases holds for M∞: (a) M∞has one end and it is asymptotic to a helicoid (in this case, Theorem 0.1 in [35] insures that M∞is a helicoid). Again, this case cannot occur as the injectivity radius function of e Enat the origin is 1 for each n∈N. (b) M∞has nonzero finite total curvature. In this case, M∞is a catenoid by the main result in [18]. (c) M∞has two limit ends. In this case, M∞is a Riemann minimal example by [33]. (G2) Lhas the structure of a limiting parking garage in the following sense: Lis a foliation of R3by parallel planes and S(L)consists of one or two lines orthogonal to the planes in L(called columns of the limiting parking garage structure), and as n→ ∞, a pair of highly sheeted multivalued graphs forms inside e Enaround each of the lines in S(L). Furthermore, if S(L)consists of two lines l, l0, then lintersects B(1),l0is at distance 1 from land the pairs of multivalued graphs inside the e Enaround different lines are oppositely handed. In fact, S(L)cannot consist of a single line; a proof of this property can be found by a direct adaptation of the second paragraph of the proof of Lemma 3.4 in [26]. 22 4.1 Finding horizontal planes Pnand “concentric” curves b Γ(n)⊂E∩Pn. Lemma 4.2 After possibly replacing Eby another end representative, there exists a sequence {Pn}n∈N∪{0}of horizontal planes with x3(Pn)< x3(Pn+1)and x3(Pn)→ ∞, such that each Pnintersects Etransversely and Pn∩Econtains a simple closed curve b Γ(n)with the following properties: 1. ∂E =b Γ(0) ⊂P0. 2. When viewed in D−{0}, each b Γ(n)with n∈Nis topologically parallel to ∂E. 3. Given n∈N, let Ωn⊂D(∗)be the finite topology subdomain whose boundary is b Γ(n)∪∂E. Then, Ωn⊂Ωn+1 for all n. 4. When viewed in R3,Ωnlies below the plane Pn. 5. Elies locally above P0along ∂E. 6. If Case (G1) occurs then: (a) For each n∈N∪ {0},b Γ(n)bounds a compact convex disk Dn⊂Pnwhose interior is disjoint from E. Furthermore, the Dnall lie in the same side of E. (b) The limit tangent plane at infinity of M∞is horizontal. 7. If Case (G2) occurs, then the planes in the limit parking garage structure are horizontal. Proof. We first claim that if Pis a horizontal plane such that ∂E ⊂ {x3< x3(P)}, then P∩E contains exactly one compact component that is nonzero in H1(D−{0})(P∩Emight contain infinitely many compact components that bound disks in D−{0}, each one containing finitely many annular ends of E). To see this, note that P∩Econtains at least one compact component that is nonzero in H1(D− {0})since ∂E lies below P, the limit end of Eis its top end and Eis connected. If P∩Econtains two compact components both nonzero in H1(D− {0}), then we can choose two of such components Γ,Γ0satisfying that Γ∪Γ0is the boundary of a compact annulus A(Γ,Γ0)⊂D−{0}such that Int(A(Γ,Γ0)) ∩x−1 3(x3(P)) does not contain components which are nonzero in H1(D−{0})and when viewed in R3,A(Γ,Γ0)∩Elocally lies above Palong Γ∪Γ0. Observe that A(Γ,Γ0)contains finitely many (annular) ends of E, each of which has nonpositive logarithmic growth. Therefore, A(Γ,Γ0)−x−1 3(−∞, x3(P)) is a parabolic surface with boundary, and x3|A(Γ,Γ0)−x−1 3(−∞,x3(P)) is a bounded nonconstant harmonic function with constant boundary values, which is impossible. This proves our claim. Assume that Case (G2) occurs for the limit of the e En.Recall that a limiting parking garage structure in R3with two oppositely handed vertical columns closely resembles geometrically and topologically a Riemann minimal example with almost horizontal flux vector and finite positive injectivity radius; we refer the reader to the paper [27] for further explanations. Let l, l0be the straight lines which are the columns of the limiting parking garage structure, and let ecn=λn(cn−pn)⊂e Enbe a connection loop for the forming parking garage structure; 23 this means that ecnis a closed curve, which approximates arbitrarily well (for nlarge enough) a path that starts at a point in the first column, travels on one level of the limiting parking garage to the second column, goes “up” one level (remember that we do not know that the columns l, l0are vertical) and then travels back again on this level “over” the previous arc until arriving at the first forming column, and then goes “down” one level until it closes up. We claim that when viewed in D−{0},cncannot bound a disk; to see this, note that if cn bounds a disk in D−{0}, then cnbounds a finite topology domain ∆nin Ewith vertical flux. Since for nlarge the flux of e Enalong ecnis arbitrarily close to a nonzero vector orthogonal to l, we conclude that l, l0are horizontal. This implies that there are points in the interior of ∆nwhose heights are strictly greater than the maximum height of cn. Since the ends of ∆n are graphical with nonpositive logarithmic growth, we find a contradiction with the maximum principle for x3|∆n. Therefore, our claim holds. We next prove that l, l0are vertical lines. Pick a plane e Pin the limiting parking garage structure, orthogonal to l, l0and for nlarge, let Pnbe a plane such that λn(Pn−pn)converges to e Pas n→ ∞, such that the height of Pndoes not coincide with the height of any planar end of E. Choose two connection loops cn, c0 n⊂Elying at different sides of Pn. Since both cn, c0 n are homologically nontrivial in D− {0}by the last paragraph, then cn, c0 nare topologically parallel in D− {0}and thus, there exists an annular domain A(cn, c0 n)⊂D− {0}bounded by cn∪c0 n. Observe that we can choose cn, c0 nso that A(cn, c0 n)contains annular ends of E (by the convex hull property). If l, l0were not vertical, then for nlarge A(cn, c0 n)∩Ewould contain interior points whose heights are strictly greater than the maximum height of cn∪c0 n, which is a contradiction as in the previous paragraph. Therefore, l, l0are vertical lines, which proves item 7 of the lemma. We continue assuming that Case (G2) occurs. By Sard’s theorem, we can assume that Pn intersects transversely E. Identifying A(cn, c0 n)∩Ewith its image minimal surface in R3, we deduce that the intersection set A(cn, c0 n)∩x−1 3(x3(Pn)) consists of a nonzero finite number of Jordan curves contained in the interior of A(cn, c0 n). By elementary separation properties, there exists at least one component b Γ(n)of A(cn, c0 n)∩x−1 3(x3(Pn)) which is topologically parallel to cnin A(cn, c0 n); in fact, b Γ(n)is unique by the arguments in the first paragraph of this proof. Thus, b Γ(n)⊂Esatisfies item 2 of the lemma. Note that the curves b Γ(n)can be chosen (after passing to a subsequence) so that the finite topology domains Ωn⊂D(∗)bounded by b Γ(n)∪∂E satisfy Ωn⊂Ωn+1 for all n, so item 3 of the lemma holds by construction. Without loss of generality, we may assume that cn⊂Ωn. Given n∈N∪ {0}and k∈N, the annulus A(b Γ(n),b Γ(n+k)) ⊂D− {0}bounded by b Γ(n)∪b Γ(n+k)satisfies that A(b Γ(n),b Γ(n+k)) ∩Eis a finitely punctured annulus and A(b Γ(n),b Γ(n+k))∩Elies below the horizontal plane at height max{x3(b Γ(n)), x3(b Γ(n+k))} (by the maximum principle applied to x3|A( b Γ(n), b Γ(n+k))∩E, since the annular ends of Ehave nonpositive logarithmic growth). As Econtains points of arbitrarily large heights because the limit end of Eis its top end, we conclude that the heights of the planes Pnare not bounded from above. After passing to a subsequence, we can assume that x3(Pn)< x3(Pn+1)and x3(Pn)→ ∞ as n→ ∞. This implies that after replacing Eby a representative of the same limit end bounded by the curve b Γ(0), we can assume that item 1 of the lemma holds provided 24 that Case (G2) occurs. Observe that the finite topology domain Ωnequals A(b Γ(0),b Γ(n)), hence item 4 holds by the last paragraph. By transversality, this implies that E−Ωnlies locally above Pnalong b Γ(n). In particular, Elies locally above P0={x3=x3(∂E)}along ∂E and item 5 of the lemma holds provided that Case (G2) occurs. Thus, the proof of Lemma 4.2 is finished if Case (G2) holds. Next assume that Case (G1) occurs for the limit of the e Enwith M∞being a Riemann minimal example. The previous arguments can be adapted to prove that: •If ecn=λn(cn−pn)⊂e Enconverges to a circle Cin the Riemann minimal example M∞, then cnwinds once around 0in D−{0}(adapt the arguments in the fourth paragraph of the present proof and use that if the flux of a Riemann minimal example is vertical, then its planar ends are not horizontal). •The limit tangent plane at infinity for M∞is vertical (adapt the arguments in the fifth paragraph of the present proof). •There exists a sequence of horizontal planes Pnsuch that {λn(Pn−pn}}nconverges to {x3=x3(C)}, and compact components b Γ(n)of E∩Pnthat are Jordan curves which, when viewed in D−{0}, wind once around 0(adapt the arguments in the sixth paragraph above). •The finite topology domain Ωn⊂D(∗)bounded by b Γ(n)∪∂E can be chosen so that Ωn⊂Ωn+1 for all n∈N, and all of the remaining properties of Lemma 4.2 hold (follow verbatim the arguments in the seventh paragraph of this proof). Finally suppose that Case (G1) occurs for the limit of the e Enwith M∞being a catenoid. Let e P, Pn⊂R3be parallel planes so that e Pintersects M∞in its waist circle eγ, and for each n Pn∩Econtains a convex Jordan curve γnsuch that {λn(γn−pn)}nconverges to eγas n→ ∞. Claim 4.3 For nsufficiently large, γnis nonzero in H1(D−{0}). Proof. Assume that γnbounds a disk ∆in D−{0}. By the convex hull property, ∆contains a finite positive number of ends of E, all of which are annular with finite total curvature and vertical (possibly zero) flux. As γnis convex, a standard application of the L´ opez-Ros deformation argument shows that ∆contains exactly one end of E. This annular end of Ehas negative logarithmic growth for nsufficiently large, as the flux of M∞along eγis nonzero. The same reason gives that M∞is a vertical catenoid, and thus, e P, Pnare horizontal planes. For nsufficiently large, consider a compact annular neighborhood A(γn)of γnin Ewith the following properties: (H1) A(γn)is bounded by two compact, convex curves in horizontal planes and the lower boundary curve of A(γn)bounds an annular end R(n)of Eof catenoidal type (with negative logarithmic growth). 25 c1 c0 A1(1) A2(1) A2(0) A1(0) L1 L0 Cyl r r JR 1 JR 2γ(not horizontal) Figure 6: The small square in the center of the figure represents the branch point of the Gauss map NRof Rwhose height is the average of the heights of the circles c0, c1. The green curves represent the intersection of R∩Cyl with the symmetry plane of R. are horizontal convex curves and γ(ε)is a Jordan curve whose image by Nis at positive spherical distance from C. We may take εsufficiently small so that N−1(C)∩R0(ε)consists of two disjoint arcs, each one joining c0(ε)to c1(ε). The complement of R0(ε)in D− {0} consists of three components, namely two annular components of which one contains ∂E and another one contains 0, and a disk component ∆. By the convex hull property, ∆contains a finite positive number of annular ends of E. Observe that we can choose an infinite sequence of pairwise disjoint domains of the type R0(ε)in E, so that the sequence collapses to the origin when viewed in D−{0}. Assertion 4.9 After replacing Eby a limit subend, every such a domain ∆contains exactly one annular end of E. Proof. Arguing by contradiction, assume that we have a sequence ∆nof such domains so that ∆ncontains at least two annular ends of E. As the limiting normal vector of Eat its annular ends is vertical and N−1(C)∩∂∆n=Ø, then a simple continuity argument gives that N−1(C)∩∆ncontains a finite positive number of components, each of which is a Jordan curve. Choose one of these Jordan curves βn⊂N−1(C)∩∆n. If for each n∈Nthere exists a point p0 n∈βnso that the sequence {IE(p0 n)}nis bounded, then Lemma 4.6 implies that after extracting a subsequence, the E−p0 nconverge smoothly with multiplicity one to R. Note that the sequence {(∆n∩E)−p0 n}nalso converges to R(because the intrinsic distance from γn(ε)to βngoes to infinity as n→ ∞). This is impossible, as every closed curve in ∆n∩Ehas vertical flux but Rdoes not have this property. Therefore, the sequence of numbers {min IE(x)|x∈βn}ngoes to ∞. In this situation, Assertion 4.8 gives a contradiction by taking for each n∈Na point xn∈βnof maximum height in R3(which exists since βnis a Jordan curve). This finishes the proof of Assertion 4.9.2 By Assertion 4.9, after replacing Eby a limit subend, we assume that every ∆-domain as defined in the description just before the statement of Assertion 4.9, contains one end of E. By the last sentence before Assertion 4.9, these ∆-domains occur in a sequence collapsing to 32 the limit end of E. By the Gauss-Bonnet formula, the total Gaussian curvature of the annulus ∆∩Eis arbitrarily small by choosing R0(ε)appropriately. Therefore, ∆∩Eis a graph over its projection into the (x1, x2)-plane, of a function with small length of its gradient (the maximum of the length of the gradient of the graphing function occurs at ∂∆). By gluing (∆ ∩E)∪R0(ε)with the two planar disks bounded by c0(ε)∪c1(ε), we obtain a piecewise smooth topological plane Π, which is properly embedded in R3. Take a maximal collection {Πn}nof such topological planes, so that Πn∩Πm=Ø if n6=m.R3−Sn∈NΠnconsists of a countable union of open components, each of which is a topological slab Sn. Given such a slab Sn, observe that the closure of N−1(C)∩Snis a compact 1-manifold with four boundary points. Therefore, N−1(C)∩Snconsists of a finite number of Jordan curves plus two arcs. We first check that for nsufficiently large, N−1(C)∩Sndoes not contain Jordan curve components. Otherwise, there exists a sequence of points p0 n∈N−1(C)∩Snwhere the tangent line to N−1(C)∩Snis horizontal. By Assertion 4.8,IE(p0 n)must be bounded. Thus, Lemma 4.6 gives that after extracting a subsequence, the E−p0 nconverge smoothly to R, which is impossible since on Rthe corresponding set N−1 R(C) = JR 1∪JR 2satisfies that the angle with the horizontal planes is bounded away from zero. Therefore, N−1(C)∩Sndoes not contain Jordan curve components that for nsufficiently large, and the same argument proves that the two compact arcs in the closure of N−1(C)∩Snmake an angle with the horizontal planes which is bounded away from zero; in particular, each of these arcs joins two boundary components of Sn. After replacing Eby a subend, we can assume that N−1(C)consists of two proper arcs J1, J2satisfying item 2 of the proposition. In particular, each Jican be parameterized by the x3-coordinate, i= 1,2. Assertion 4.10 Given δ > 0, there exists x3,0=x3,0(δ)∈Rsuch that if x3≥x3,0, then for i= 1,2it holds IE(Ji(x3)) ≤δ+ lim sup x∈JR 1∪JR 2 IR(x). Proof. Arguing by contradiction, suppose that the assertion fails. Then, there exists δ > 0and a sequence of heights tn→ ∞so that IE(Ji(tn)) > δ +c, where c= lim supx∈JR 1∪JR 2IR(x). After passing to a subsequence, we can assume that given n∈N, there exists t0 n∈(tn, tn+1)so that the related point Ji(t0 n)lies in a region of the form R0(ε)where Eis ε-close to a compact portion of R. Then, after taking εmuch smaller than δ, we can assume that IE(Ji(t0 n)) ≤δ 2+c. By continuity of IE◦Ji, there exists t00 n∈(tn, t0 n]such that IE(Ji(t00 n)) = δ 2+cfor each n∈N. Applying Lemma 4.6 to pn:= Ji(t00 n)we deduce that the E−Ji(t00 n)converge (after extracting a subsequence) smoothly to R, which is impossible since the value of the injectivity radius of E−Ji(t00 n)at the origin is δ 2+cand the injectivity radius is a continuous function with respect to smooth limits (see e.g. Ehrlich [13] and Sakai [42]). Now the assertion is proved. 2 Finally, item 1 of Proposition 4.7 follows directly from Assertion 4.10. Item 2 of Proposition 4.7 follows from Assertion 4.10, Lemma 4.6 and the fact that the unit tangent vector along the curves JR 1∪JR 2makes an angle with the horizontal planes which is bounded away from zero. This completes the proof of the proposition. 2 33 A direct consequence of Lemma 4.6 and Assertion 4.10 is that for every divergent sequence of points p0 n∈J1∪J2, the surfaces E−p0 nconverge smoothly to Rafter passing to a subsequence. This property together with Assertion 4.9 imply that after replacing Eby a subend, Econsists of an infinite number of noncompact pieces Mn, each of which is has the topology of a pair of paints with a point removed (this puncture is one annular end of E), and the two compact boundary components of c0,n, c1,n of Mncan be taken arbitrarily close to translated copies of the horizontal circles c0, c1⊂ R defined in the paragraph just after the proof of Assertion 4.8. Furthermore, c1,n =c0,n+1 and Mn∩Mn+1 =c1,n for all n∈N. We next explain why Theorem 1.6 holds in the Case (G1) when M∞is a Riemann minimal example. The main properness statement of Theorem 1.6 was proven in Section 3. Items 1, 2 of Theorem 1.6 were proven in the second paragraph of this section 4. Item 3 of Theorem 1.6 follows from Lemma 4.2 and Corollary 4.5. In particular, items 1, 2, 3 of Theorem 1.6 also hold in the Case (G1) when M∞is a vertical catenoid. Assume from now on that Case (G1) occurs and M∞is a Riemann minimal example. Item 4 of Theorem 1.6 is a consequence of the last paragraph. The same description of Eas a union of domains Mnimplies that the Gaussian curvature of Eis bounded, which is item 5 of Theorem 1.6. The next proposition completes the proof of Theorem 1.6 in the Case (G1) when M∞is a Riemann minimal example. Proposition 4.11 If Case (G1) occurs and M∞is a Riemann minimal example, then item 6 of Theorem 1.6 holds. Proof. Suppose that the proposition fails. As each of the annular ends of Ehas finite total curvature, Eis conformally diffeomorphic to b D=D−{x∈D| |x| ≤ a}for some a∈(0,1), with a countable discrete set of points {en}n∈Nremoved and where |en| & aas n→ ∞. By the above decomposition of Eas a countable union of regions Mn, there exists δ > 0 and a sequence fn:S1×[0, δ]→E(here S1is the unit circle) of conformal embeddings with fn(S1×[0, δ]) being arbitrarily close to a region R0(ε)of ‘Riemann type’ to which one attaches an annular end of E(that might have negative logarithmic growth, arbitrarily close to zero). Observe that the fnhave pairwise disjoint images in R3for different values of n. Consider on b Dthe usual flat metric g0. Next we will show that the g0-area of fn(S1×[0, δ]) is at least 2πa2δ, which gives the desired contradiction since the g0-area of b Dis finite and we have an infinite number of such pairwise disjoint embeddings fnin b D. To compute the g0-area of fn(S1×[0, δ]), we will apply the coarea formula to the smooth function hn:fn(S1×[0, δ]) →Rthat satisfies (hn◦fn)(θ, t) = t, for all (θ, t)∈S1×[0, δ].(8) Thus, Area(fn(S1×[0, δ]), g0) = Zδ 0 Zh−1 n(t) dst |∇0hn|!dt, (9) where dst,|∇0hn|denote respectively the length element of the simple closed curve h−1 n(t) = fn(S1×{t})and the gradient of hn, both computed with respect to g0. Since fnis a conformal 34 diffeomorphism onto its image endowed with g0, we deduce that v:= 1 ∂fn ∂t (θ, t) ∂fn ∂t (θ, t)(10) is a unit normal vector to the curve fn(S1×{t})at the point fn(θ, t). Hence, (8) and (10) give |∇0hn|(fn(θ, t)) = (dhn)fn(θ,t)(v) = 1 ∂fn ∂t (θ, t) , which implies that the right-hand-side of (9) equals Zδ 0 Zfn(S1×{t}) ∂fn ∂t dst!dt =Zδ 0 ZS1×{t} ∂fn ∂t  ∂fn ∂θ dθ!dt. (11) Using again the conformality of fnin the right-hand-side of (11) and the Cauchy-Schwarz inequality, we obtain Area(fn(S1×[0, δ]), g0) = Zδ 0 ZS1×{t} ∂fn ∂θ  2 dθ!dt ≥1 2πZδ 0 ZS1×{t} ∂fn ∂θ dθ!2 dt =1 2πZδ 0 [length(fn(S1×{t}))]2dt (?) ≥1 2πZδ 0 (2πa)2dt = 2πa2δ, where in (?)we have used that fn(S1× {0})is a loop in b Dthat is parallel to ∂b D. This completes the proof of the proposition. 2 4.3 Analysis of the Case (G1) when M∞is a catenoid. We will devote this section to prove Theorem 1.6 provided that Case (G1) occurs and that the limit surface M∞of the surfaces e Engiven by (2) is a vertical catenoid. Without loss of generality, we will assume that the waist circle of M∞is the unit circle in the (x1, x2)-plane. Recall the following properties demonstrated above for each n∈N: (J1) There exists a horizontal plane Pnso that Pn∩Econtains a convex Jordan curve b Γ(n) and the λn(b Γ(n)−pn)converge as n→ ∞ to the waist circle eγof M∞. Moreover, the heights of Pndiverge increasingly to ∞. (J2) b Γ(0) = ∂E,b Γ(n)is topologically parallel to ∂E in D−{0}(items 1 and 2 of Lemma 4.2) and b Γ(n)is the unique compact component of Pn∩Ethat is topologically parallel to ∂E in D−{0}(claim in the first paragraph of the proof of Lemma 4.2). (J3) When viewed in D(∗),b Γ(n)bounds a noncompact domain E(b Γ(n)) which is an end representative of the limit end of E; we take E(b Γ(n)) as the closure of the component of E−b Γ(n)such that E(b Γ(n)) ∩∂E =Ø. 35 (J4) When viewed in R3,b Γ(n)bounds a compact convex disk Dn⊂Pnwhose interior is disjoint from E, and the Dnall lie in the same side of E(item (6a) of Lemma 4.2). We will denote by Wthe closure of the component of R3−(E∪D0)that contains Dnfor n≥1. (J5) F(b Γ(n)) = FE−2πβne3and F(b Γ(n))H= (FE)His not zero (items 1 and 2 of Corollary 4.5). (J6) VE=∞,β∞=Pnβn=−∞and λn→0as n→ ∞ (items 4 and 6 of Corollary 4.5). In particular, the annular ends of Eall have strictly negative logarithmic growths (because if one of these ends were asymptotic to a plane, then all annular ends of Eabove this last one would be planar as well, and F(b Γ(n))Vwould then be independent of n, which contradicts that β∞=−∞after taking vertical components in the first formula of (J5)). (J7) {IE(p0 n)}nis unbounded for every divergent sequence {p0 n}n⊂E(Lemma 4.6). Proposition 4.12 Given ε > 0small, there exist compact annular subdomains ∆n= ∆n(ε)⊂ Ebounded by horizontal convex curves, such that for nsufficiently large: 1. There exist numbers λ0 n>0converging to zero and points p0 n∈R3such that the Hausdorff distance between λ0 n(∆n−p0 n)and M∞(ε) = {x∈M∞:|x3| ≤ 1/ε}is less than ε, and λ0 n(∆n−p0 n)can be written as a normal graph over its projection to M∞with C2-norm less than εand the boundary curves of λ0 n(∆n−p0 n)are contained in the planes {x3=±1/ε}. 2. For all n∈N, the boundary curves of ∆nare topologically parallel to ∂E in D−{0}. Therefore, we may assume that the ∆nare ordered so that for each n,∆nis contained in the component of D(∗)−∆n+1 that contains ∂E. 3. The closed horizontal slab in R3that contains ∆nis strictly below the one that contains ∆n+kfor all n, k ∈N,k6= 0. 4. Except for a finite number of components, each component Ωof E−[ n∈N ∆nis topologically a plane with two disks removed, and Ωis the graph of a function udefined over the projection of Ωto the (x1, x2)-plane, with |∇u|<1. 5. The Gaussian curvature KEof Eis asymptotically zero. Proof. Items 1 and 2 follow from the facts that the sequence {e En}ndefined by (2) converges smoothly on compact subsets of R3with multiplicity 1 to the vertical catenoid M∞, that λn→ 0, and that the convex horizontal curves b Γ(n)defined in (J1) are topologically parallel to ∂E in D−{0}. We next prove item 3. Let A(b Γ(n),b Γ(n+ 1)) be the subdomain of Ebounded by b Γ(n)∪ b Γ(n+1). Since the ends of A(b Γ(n),b Γ(n+1)) have negative logarithmic growths and {x3(Pn)}n is increasing, the maximum principle applied to the function x3|A( b Γ(n), b Γ(n+1)) implies that 36 A(b Γ(n),b Γ(n+ 1)) is contained in the halfspace {x3≤x3(b Γ(n+ 1))}. As A(b Γ(n−1),b Γ(n)) is contained in the halfspace {x3≤x3(b Γ(n))}and A(b Γ(n−1),b Γ(n)) ∩A(b Γ(n),b Γ(n+ 1)) = b Γ(n), then we conclude that A(b Γ(n),b Γ(n+ 1)) locally lies above Pnalong b Γ(n). Observe that A(b Γ(n),b Γ(n+ 1)) contains the portion ∆+ nof ∆nthat lies above Pn, and it also contains the portion ∆− n+1 of ∆n+1 that lies below Pn+1. By the maximum principle applied to x3|A( b Γ(n), b Γ(n+1))−[∆+ n∪∆− n+1], we deduce that the lower boundary curve of ∆n+1 lies strictly above the upper boundary curve of ∆n, which implies that item 3 holds. Next we prove item 4 of the proposition. For ε > 0small and fixed, choose a maximal collection of pairwise disjoint domains {∆n}n∈N∪{0}⊂Ewhich satisfy items 1, 2 and 3. After replacing Eby a subend representative of the limit end, we may assume that ∂E is thre bottom boundary component of ∆0. We will first show that if a component Ωof E−Sn∈N∪{0}∆nis topologically a plane with two disks removed, then Ωis the graph of a function udefined over the projection of Ωto the (x1, x2)-plane, with |∇u|<1. To see this, note that if ε > 0is sufficiently small, the total geodesic curvature of Ealong each of the two components of ∂Ωis arbitrarily close to −2π. As we are assuming that Ωhas exactly one end and this end has finite total curvature, then the Gauss-Bonnet formula gives that Ωhas arbitrarily small total Gaussian curvature by taking ε sufficiently small. Therefore, the Gaussian image of Ωlies in a small neighborhood of one of the poles, say the north pole, of the unit sphere. Hence, the projection of Ωto the (x1, x2)- plane is a proper submersion which is injective on each of the two boundary components of Ω. In this setting, a straightforward covering space type argument implies that Ωis a graph of a smooth function udefined over the projection of Ωto the (x1, x2)-plane. The fact that |∇u|<1follows from the fact that the Gaussian image of Ωlies in a small neighborhood of the north pole. Therefore, to prove item 4 we must show that except for a finite number of components of E−Sn∈N∪{0}∆n, all these components have the topology of a plane minus two disks. Observe that item 2 of this proposition implies that every component of E−Sn∈N∪{0}∆nis a planar domain with two boundary components and a finite number of annular ends with negative logarithmic growth. Let {Ωn}n∈Nbe the collection of components of E−Sn∈N∪{0}∆n. Enumerate these components so that Ωnis the component of E−Sn∈N∪{0}∆nwith boundary components αn= Ωn∩∆n−1, βn= Ωn∩∆n.(12) Fix for each n∈Na dilation fn:R3→R3so that the Hausdorff distance between fn(∆n) and M∞(ε)is minimized, and so, by the definition of ∆n, this Hausdorff distance is less than ε. Lemma 4.13 After passing to a subsequence, the surfaces fn(Ωn)converge with multiplicity one to the representative M∞∩{x3≤ −1/ε}of the bottom end of M∞. In fact, the surfaces fn(E)converge smoothly on compact subsets of R3to M∞. Proof. We first show that if the sequence of curves {fn(αn)}ndiverges to infinity in R3, then the lemma holds. Following the notation in property (J3) above, we denote by E(αn)the 37 closure of the component of E−αnsuch that E(αn)∩∂E =Ø. By construction, after passing to a subsequence, we may assume that the curves fn(βn)converge in the C2-norm to a convex horizontal curve b β. Take a divergent sequence {Rn}nof positive numbers so that for each n, the boundary fn(αn)of fn(E(αn)) lies outside of the closed ball B(Rn)centered at the origin. Then, item 3 of Theorem 2.2 in [26] applied to the sequence of compact minimal surfaces {fn(E(αn)) ∩B(Rn)}nimplies that after extracting a subsequence, the fn(E(αn)) ∩B(Rn) converge smoothly on compact subsets of R3with multiplicity one to a connected, properly embedded, nonflat minimal surface c M∞of genus zero, that is either a catenoid, a helicoid or a Riemann minimal example. Clearly, c M∞contains the curve b β.c M∞cannot be a helicoid since c M∞has nonzero vertical flux along b β; the same argument shows that either c M∞is a vertical catenoid or a Riemann minimal example with vertical flux. Our earlier arguments imply that c M∞must be a vertical catenoid. Since the limit set of the sequence {fn(E(αn)) ∩B(Rn)}n equals the limit set of {fn(E)}i(this follows from the properties that fn(E−E(αn)) lies in the halfspace {x3≤x3(fn(αn))}nand x3(fn(αn)) → −∞as n→ ∞), then we conclude the Lemma in the special case that the fn(αn)diverge to infinity in R3. We next divide the proof into two parts: in the first one, we will prove the lemma assuming that, after choosing a subsequence, Ωncontains just one annular end for every n. In the second part we will suppose that, after choosing a subsequence, Ωncontains more than one annular end for every n. Assume that Ωncontains just one annular end for every n. We will demonstrate that the curves fn(αn)diverge to infinity in R3. Arguing by contradiction, assume that after choosing a subsequence, fn(αn)lies in a compact set of R3independently of n∈N. Recall that the logarithmic growths of the annular ends of Eare bounded (between the negative logarithmic growth of the lowest end of Eand zero), and that the dilation fnhas homothetic factor going to zero as n→ ∞. Therefore, the logarithmic growths of the unique annular end of fn(Ωn) is arbitrarily small in absolute value for nsufficiently large. Since the flux of fn(Ωn)along fn(βn)is converging to the nonzero flux of M∞along b βand the flux of fn(Ωn)along its annular end is arbitrarily small, then the divergence theorem implies that the flux of fn(Ωn)along fn(αn)converges to the negative of the flux of M∞along b β, that is nonzero. This property and the fact that the convex planar curve fn(αn)lies in a compact set independent of n, imply that the fn(αn)converge (after passing to a subsequence) to a convex, horizontal planar curve bαas n→ ∞. Also recall that fn(Ωn)is a graph over its projection to the (x1, x2)-plane, by the fourth paragraph in the proof of Proposition 4.12. Therefore, curvature estimates for stable minimal surfaces imply that the fn(Ωn)converge to a minimal graph over the complement in the plane {z= 0}of the two disks bounded by Π(bα),Π(b β), where Π(x, y, z) = (x, y, 0). As this minimal graph has vertical flux and two convex boundary curves, a standard application of the L´ opez-Ros deformation argument leads to contradiction. This contradiction shows that the fn(αn)diverge to infinity in R3. By the discussion in the first paragraph of this proof, we now conclude that Lemma 4.13 holds if Ωnhas one end for every n(after choosing a subsequence). Next assume that Ωnhas always at least two ends. Again by the discussion in the first paragraph of the proof of Lemma 4.13, it remains to show that the curves fn(αn)diverge to infinity in R3. Assume this last property fails to hold. Since the diameter of the sets fn(αn)are 38 uniformly bounded (because the diameter of fn(∆n−1)is bounded as the homothetic factor of the dilation fnis going to zero and the diameter of fn−1(∆n−1)is comparable to the one of M∞(ε)), we may assume from this point on that the curves fn(αn)all lie in a fixed bounded subset of R3. This bounded set must lie below the plane {x3=−1}if εis chosen sufficient small (by the already proven item 3 of Proposition 4.12). We will find the desired contradiction by analyzing each of the following two mutually exclusive situations (after passing to a subsequence): (K1) The diameters of the curves fn(αn)are not bounded away from zero. (K2) The diameters of the curves fn(αn)are bounded away from zero. Suppose that Case (K1) holds. Then, after choosing a subsequence, we may assume that the curves fn(αn)converge to a point p∈R3that lies below the plane {x3=−1}. Consider the sequence of compact, embedded, minimal planar domains {fn(E(αn)) ∩B(n)}n. We claim that the [fn(E(αn))∩B(n)]−{p}form a locally simply connected sequence of minimal planar domains in R3− {p}. Otherwise, our previous arguments show that we can produce, after blowing-up by topology, a new limit of dilations of the [fn(E(αn)) ∩B(n)] −{p}which is a vertical catenoid. This means that after extracting a subsequence and for nsufficiently large, [fn(E(αn)) ∩B(n)] −{p}contains a compact subdomain Cnwhich is arbitrarily close to a homothetically shrunk copy of a large compact region of a vertical catenoid, where the homothetic factor can be taken arbitrarily small. Note that Cncannot lie in fn(Ωn)because this contradicts the maximality of the family {∆m}m. Since fn(∆n)is ε-close to M∞(ε), we deduce that Cnmust lie in fn(E(αn+1)). To see that this is impossible, first observe that for n sufficiently large, the generator of the homology group H1(Cn)of Cnis topologically parallel to fn(βn)modulo annular ends of fn(E)(adapt the arguments as in the proof of Claim 4.3). As the vertical component of the flux vector of fn(∆n)along fn(βn)is larger than some positive number in absolute value (namely, one half of the vertical flux of M∞) and the annular ends of fn(E)all have negative logarithmic growths, we deduce from the divergence theorem that the vertical component of the flux vector of fn(Cn)is positive and bounded away from zero (see the last paragraph of the proof of Corollary 4.5), which contradicts that the length of a generator of H1(Cn)tends to zero as n→ ∞. Therefore, the sequence [fn(E(αn)) ∩B(n)] − {p}is locally simply connected sequence in R3− {p}. In fact, this argument shows that for all n∈Nand given a regular neighborhood Un(δ)of the boundary of fn(E(αn)) ∩B(n)in fn(E(αn)) ∩B(n)with radius δ > 0, the restriction of the injectivity radius function of fn(E) to [fn(E(αn)) ∩B(n)] −Un(δ)is uniformly bounded away from zero (independently of n). In this setting, item 3 of Theorem 2.2 in [26] ensures that after passing to a subsequence, the surfaces [fn(E(αn))∩B(n)]−{p}converge to a minimal lamination Lof R3−{p}whose closure Lin R3consists of a single leaf which is a properly embedded minimal surface L1of genus zero that is either a helicoid, a catenoid or a Riemann minimal example. Furthermore, the convergence of the [fn(E(αn))∩B(n)]−{p}to L1is smooth on compact sets of R3−{p}. Our previous arguments show that L1is the vertical catenoid M∞. As pis a point in L1, then M∞must pass through p. We next analyze the intersection of fn(E(αn)) with a ball B(p, δ) 39 of small radius δ > 0so that B(p, δ)∩M∞is a graphical disk with boundary Γ∞. For n large, we can assume that fn(αn)⊂B(p, δ). Since fn(E(αn)) is a properly embedded surface of genus zero and fn(E(αn)) ∩∂B(p, δ)consists of a single curve Γnsuch that {Γn}n→ Γ∞, then we conclude that fn(αn)∪Γnbounds a compact annulus in fn(E(αn)); in fact, fn(E(αn)) ∩B(p, δ)is this annulus. We now arrive at the desired contradiction as follows. Consider a horizontal plane Πstrictly below the height of p. Since M∞is the smooth limit of the [fn(E(αn)) ∩B(n)] − {p}away from p, we conclude that the horizontal circle M∞∩Πis arbitrarily close to a simple closed, planar convex curve cn⊂fn(E(αn)). Observe that cncan be joined to both fn(αn)and fn(βn)by arcs that do not intersect fn(αn)∪fn(βn)∪cnexcept at their extrema. Therefore, fn(αn)∪fn(βn)∪cnis the boundary of a compact planar domain in fn(E(αn)). Since αn and βnare both nontrivial in D− {0}, then cnmust bound a disk in D− {0}and therefore, cnbounds in Ea punctured disk Tnwith the number of punctures being positive and finite (depending on n) by the convex hull property. Recall that the logarithmic growths of the annular ends of fn(En)are arbitrarily small in absolute value for nsufficiently large. As the flux of fn(E(αn)) along cnis bounded away from zero, we conclude that the number of punctures in Tnis unbounded as n→ ∞. Since fn(Tn)lies below the plane Π,cn=∂[fn(Tn)] is a convex horizontal curve inside Πand the flux of any closed curve in fn(Tn)is vertical, then the L´ opez-Ros deformation argument implies that Tncontains just one puncture, which is a contradiction for nlarge. This contradiction proves that Case (K1) does not occur. Finally suppose that Case (K2) occurs. This assumption implies that after passing to a subsequence, the following properties hold. (L1) The surfaces fn(∆n−1)converge to a compact minimal annulus ∆∞bounded by two horizontal simple closed convex curves, and ∆∞is close in the Hausdorff distance to a compact piece of a vertical catenoid. Similarly, fn(∆n)converge to a compact minimal annulus ∆∞bounded by two horizontal simple closed convex curves, and ∆∞is ε-close in the Hausdorff distance to M∞(ε). (L2) The curves fn(αn)converge to the top boundary component bαof ∆∞. (L3) For all n∈N, the restriction of the injectivity radius function of fn(E)to fn(E(αn)) ∩ B(n)is uniformly bounded away from zero (independently of n; this is a consequence of the arguments in the first paragraph of the proof of Case (K1)). We now divide the argument of why Case (K2) leads to contradiction into two subcases, depending on whether or not the sequence {fn(Ωn)}nhas locally bounded second fundamental form in R3. First suppose that {fn(Ωn)}nhas locally bounded second fundamental form in R3. By the arguments in the proof of Lemma 1.1 in [35], after passing to a subsequence, the surfaces fn(Ωn)converge to a minimal lamination L1of R3−(bα∪b β). In fact, property (L1) above together with the definition of fnimply that {fn(Int(Ωn∪∆n−1∪∆n))}nconverges to a minimal lamination L2of R3−(bα1∪b β1), where bα1=∂∆∞−bα, b β1=∂∆∞−b β, 40 and L2contains the interior of both ∆∞,∆∞as portions of its leaves. We will call L(∆∞) (resp. L(∆∞)) the leaf of L2that contains the interior of ∆∞(resp. of ∆∞). Note that L(∆∞)might coincide with L(∆∞). Also observe that neither L(∆∞)nor L(∆∞)are stable, as both ∆∞,∆∞can be assumed to be unstable by choosing εin the statement of Proposition 4.12 sufficiently small. As L(∆∞), L(∆∞)are not stable, Theorem 1 in [32] implies that L(∆∞), L(∆∞)are not limit leaves of L2. Also note that every limit leaf of L1is contained in a limit leaf of L2, and so, limit leaves of L1are complete stable minimal surfaces, which are planes. In particular, limit leaves of L1and of L2are the same. This implies that the closure in R3of each nonflat leaf of L2is proper in R3, in a halfspace or in a slab with boundary being limit leaves of L1. In particular, the following surfaces with compact boundary are proper in R3, proper in an open halfspace or proper in an open slab: (L(∆∞)∪bα1, L(∆∞)∪b β1if L(∆∞)6=L(∆∞), L(∆∞)∪bα1∪b β1if L(∆∞) = L(∆∞). Suppose that L(∆∞)6=L(∆∞). In this setting, property (L3) above and the intrinsic version of the one-sided curvature estimates by Colding and Minicozzi (Corollary 0.8 in [9]) imply that L(∆∞)∪bα1has bounded Gaussian curvature in any small regular neighborhood of the limit set of L(∆∞)∪bα1(see Lemma 1.2 in [35] for a similar argument using the extrinsic version of the one-sided curvature estimates by Colding and Minicozzi). Therefore, L(∆∞)∪bα1is proper in R3, and the same holds for L(∆∞)∪b β1by similar arguments. In the case that L(∆∞) = L(∆∞), the same reasoning gives that L(∆∞)∪bα1∪b β1is proper in R3. Observe that L(∆∞)has genus zero and one or two boundary curves, each of which bounds an open convex horizontal disk disjoint from L(∆∞)(this follows from the arguments in the proof of Assertion 3.13). If L(∆∞)has one boundary curve, then we contradict the Halfspace Theorem, as L(∆∞)lies in the halfspace {x3≤0}(because fn(Ωn)has the same property for all n) and L(∆∞)contains interior points with heights strictly greater than its boundary curve. Therefore, L(∆∞)has two boundary curves (equivalently, L(∆∞) = L(∆∞)). By the convex hull property, L(∆∞)is noncompact. As L(∆∞)−∆∞is contained in {x3≤0}, then L(∆∞)has horizontal limit tangent plane at infinity. Since both αn, βnhave the same horizontal component of their fluxes, and the homothetic factors λnin (2) tend to zero, then we deduce that the fluxes of L(∆∞)along its boundary curves are vertical. This implies that L(∆∞)has vertical flux, since it has genus zero. Therefore, L(∆∞)cannot have a finite positive number of ends by the L´ opez-Ros deformation argument. The previous paragraph implies that L(∆∞)has infinitely many ends. Let Dbe a positive number such that the boundary of L(∆∞)is contained in the ball B(D)of radius Dcentered at the origin. We claim that for every sequence {µn}nof positive numbers going to zero, the restriction to µn[L(∆∞)−B(2D)] of the injectivity radius function of µnL(∆∞)is greater that some positive constant (independent of n) times the distance to the origin. Otherwise, there exists a sequence of points xn∈µn[L(∆∞)−B(2D)] such that IµnL(∆∞)(xn) |xn|→0as n→ ∞, 41 of domains ∆nand Ωnas given in Proposition 4.12 implies that item 4 of Theorem 1.6 holds. Item 5 of the same theorem follows from item 5 of Proposition 4.12. Finally, the arguments in the proof of Proposition 4.11 can be easily adapted to the current situation, with the only change of the annular regions of “Riemann type” by similar annular regions of “catenoid type”, namely regions of the type of the compact annuli ∆n= ∆n(ε)that appear in Proposition 4.12, each of which contains the image of a conformal embedding fn(S1×[0, δ]) for some δ > 0 independent of n(here we are using the notation in the proof of Proposition 4.11). This finishes the proof of Proposition 4.16.2 5 The proof of Theorem 1.3. Suppose M⊂R3is a complete, embedded minimal surface with finite genus, an infinite number of ends and compact boundary. We first check that Mhas at most two simple limit ends. Arguing by contradiction, suppose Mhas at least three simple limit ends, say e1,e2,e3. By Theorem 1.6, we can choose three pairwise disjoint, properly embedded representatives E1, E2, E3⊂M, representing e1,e2,e3 respectively, such that each one satisfies, after a possible rotation, the conclusions of Theorem 1.6. Embeddedness of Mimplies that after a rotation, the annular ends of E1, E2, E3 may be assumed to be asymptotic to ends of horizontal planes and catenoids with vertical axes. Furthermore, after a possible reindexing, we may assume that the ends E1, E2are simple top limit ends, that ∂E1is a simple closed curve in the (x1, x2)-plane and that ∂E2has constant positive x3-coordinate. Let DE1⊂ {x3= 0}be the planar disk with ∂DE1=∂E1and let X1be the closure of the component of R3−(E1∪DE1)that lies above DE1locally near DE1. Similarly, we can define a horizontal disk DE2with ∂DE2=∂E2and the related closed component X2of R3−(E2∪DE2)above DE2. An elementary topological analysis applied to the topological picture of E1and E2given in item 4 of Theorem 1.6 shows, after possibly reindexing E1and E2and replacing E1and E2by representing subends, that DE2∩E16=Ø and X2contains a representative E0 1⊂E1 of the limit end of E1with ∂E0 1⊂DE2⊂∂X2. Let X3be the closure of the component of X2−E0 1which has ∂E2in its boundary. The piecewise smooth surface ∂X3is a good barrier for solving least-area problems in X3(Meeks and Yau [40]), see Figure 7. Let e E2be a noncompact, properly embedded surface of least-area in X3with ∂e E2= ∂E2⊂∂X3. By a result of Fischer-Colbrie [15], the orientable, stable minimal surface e E2has finite total curvature. Since e E2is contained in X2, it must have a finite number of ends, all of which are annuli and which are parallel to the planar and catenoidal ends of E2. Since points of e E2near DE2have x3-coordinates which are larger than the constant value x3(DE2),e E2must have a highest end which has positive logarithmic growth by the maximum principle applied to the harmonic function x3|e E2. Hence, e E2has a catenoid-type end representative Fof positive logarithmic growth. Since the annular ends of E1have nonpositive logarithmic growth, then 48 Figure 7: The area-minimizing surface e E2is trapped between two simple top limit ends E1, E2. none of the annular ends of E1lie above F. This implies that E1lies below the region of R3bounded by the union of Fand a horizontal disk with boundary in F. Since E1also lies above some catenoid end of negative logarithmic growth, the results in [12] imply that E1has quadratic area growth. By the monotonicity formula, each annular end in E1contributes with at least π 2R2to the area growth of E1in each ball B(R)for R > 0large. Hence, E1has a finite number of ends. This contradiction shows that Mcannot have more than two simple limit ends, which is item 1 of Theorem 1.3. Next we prove item 2 of Theorem 1.3. If Mis properly embedded in R3, then the results in [12] imply Mhas one or two limit ends, which are the top and/or bottom ends in the ordering of the ends of M. On the other hand, if Mhas one or two limit ends, then these limit ends are simple limit ends, and so, these limit ends have proper representatives by Theorem 1.6. The remaining finite number of ends of Mare then annuli, each of which is proper (see Theorem 3.3 in Section 4). Hence, Mis proper, which proves item 2 in Theorem 1.3. Concerning item 3, suppose now that Mhas a countable number of limit ends. A result proven in pages 288, 289 of [24] states that the space of ends of Membeds topologically as a totally disconnected, closed subset Aof the closed unit interval I= [0,1]. Since the set of limit points LAof Ais a closed countable subspace of the metric space I(and hence LAis complete), Baire’s theorem implies that LAcontains a countable dense set of isolated points (see Lemma 5.1 below). In particular, if LAhas at least three points, then Mhas at least three simple limit ends. Since Mcannot have more than two simple limit ends by item 1 of Theorem 1.3, then LAconsists of one or two points, and so Mhas one or two limit ends, each of which is a simple limit end. Hence, part 3-A of Theorem 1.3 holds. As Mhas at most two limit ends, then item 2 of Theorem 1.3 implies that Mis properly embedded in R3, which is part 3-B. If Mhas exactly two limit ends, then these limit ends are simple. By the proof of item 1 of Theorem 1.3, we deduce that after a rotation of M, these simple limit ends have representatives 49 E1, E2, where E1is a top limit end of Mand E2is a bottom limit end of M. By item 1 of Theorem 1.6, the annular ends of E1have nonpositive logarithmic growths and the annular ends of E2have nonnegative logarithmic growths. Thus, the embeddedness of Mimplies that all the annular ends of Mmust have zero logarithmic growth, which means that they are planar, and item 3-C is proved. Now assume ∂M =Ø. Since Mhas finite genus, then the main result in [31] insures that Mhas two limit ends and is recurrent for Brownian motion, which is part 3-D. We finally prove item 3-E of Theorem 1.3. Assume ∂M 6=Ø. If the annular ends of Mare horizontal and planar, there exists a horizontal plane Pthat intersects Mtransversely in a finite number of simple closed curves, and Pcan be chosen to lie above ∂M. Hence, the closure Σ of each component of M−Pis a properly embedded minimal surface with compact boundary and Σis contained in a closed halfspace of R3. Theorem 3.1 in [12] implies that such a Σis a parabolic surface with boundary. Since there are a finite number of such closed components Σ, and the union of these components along related compact boundary components is M, we conclude that ∂M has full harmonic measure, and so item 3-E holds provided that all of the annular ends of Mare horizontal and planar. If there exists an annular end with nonzero (say negative) logarithmic growth, then this end is asymptotic to the end of a negative half catenoid, and so, there exists a horizontal plane P whose intersection with this catenoidal end is an almost circle, and the end has a representative Ewith ∂E ⊂Psuch that Eis graphical over the outside of the open planar disk D⊂Pwhose boundary is ∂E. We may assume that Pis at height zero. The complement of the topological plane E∪Din Mconsists of several components, each one with nonempty boundary contained in M∩D. Since Mis proper, then M∩Dis compact. In particular, M−(E∪D)has a finite number of components. Let Σ1,...,Σkbe the components of M−(E∪D)which lie below E∪D. For each i= 1, . . . , k, the surface with boundary Σiis parabolic, since its third coordinate function is a proper negative harmonic function. By items 3-A and 3-C, the surface Mhas exactly one limit end. Since a limit end of a properly embedded minimal surface in R3 cannot lie below a catenoidal end of negative logarithmic growth (see Lemma 3.6 in [12]), then the limit end of Mhas a representative of genus zero ETwhich lies above E∪D. In particular, the limit end of Mis its top end. By item 6 of Theorem 1.6, the representative ETis parabolic. Let Ωbe the closure of one of the (finitely many) components of M−(ET∪Σ1∪. . . ∪Σk). Since Ωhas a finite number of ends, each of which is asymptotic to an end of a plane or half catenoid, then Ωhas quadratic area growth. Therefore, Ωis also a parabolic surface with boundary. As Mis a finite union of parabolic surfaces with boundary along their common compact boundaries, we deduce that Mhas full harmonic measure on its boundary. This finishes the proof of Theorem 1.3.2 For the sake of completeness, we prove the following elementary fact which was needed in the above proof. Lemma 5.1 Suppose Xis a complete countable metric space, L(X)⊂Xis the set of limit points of Xand S(X) = X−L(X)is the open set of nonlimit points of X. Then: 1. S(X)is dense in X. 50 2. L(X)is a complete countable metric space, and so, its set S(L(X)) of isolated points is dense in L(X). Proof. Let L(X) = {p1, ..., pn, ...}be a listing, possibly finite or empty, of the set of limit points of X. If L(X) = Ø, then S(X) = S(X) = X, and so, item 1 holds. Otherwise, consider the subsets Xn=X−{p1, ..., pn}and note that each Xnis an open dense subset of X. The intersection of this countable collection of sets is equal to S(X)and must be dense in Xby Baire’s theorem. Hence, S(X)is dense in X, which proves item 1 in the lemma. Since S(X)is an open set and Xis a complete countable metric space, then L(X) = X−S(X)is a closed countable set which is complete in the induced metric. Hence, by item 1, S(L(X)) is dense in L(X).2 6 The proof of Corollary 1.8. This last section is devoted to the following result, which has Corollary 1.8 stated in the Introduction as a special case. Corollary 6.1 Suppose that M⊂R3is a connected properly embedded minimal surface with compact boundary and a limit end of genus zero. Then Mis recurrent for Brownian motion if its boundary is empty, and otherwise its boundary has full harmonic measure. Proof. Suppose for the moment that the corollary holds when the surface Mhas nonempty boundary. Then, in the special case that the boundary of Mis empty, consider a compact disk D⊂Mand note that M−Int(D)has full harmonic measure by assumption, which implies that Mis recurrent for Brownian motion. Thus, it suffices to prove the corollary in the special case that Mhas nonempty boundary. Assume now that ∂M 6=Ø. Let b E⊂Mbe an end representative of a limit end of M of genus zero. Since b Eis proper in R3with compact boundary, then item 2 of Theorem 1.3 implies that b Ehas one or two simple limit ends. Let E⊂b Ebe an end representative of a simple limit end of Mof genus zero. After a fixed rotation of Mand a replacement of E by a subend representative of its limit end, we may assume that Esatisfies the conclusions of Theorem 1.6, and ∂M ⊂ {x3<0}. We claim that there exist a pair of horizontal open disks D1, D2⊂R3−Ewith the following properties. (N1) ∂Di⊂E,i= 1,2, and 0≤x3(D1)< x3(D2). (N2) D1∩E=Ø and if we denote by X1the closure of the mean convex region of R3−(E∪ D1), then D2⊂R3−X1. In particular, D2∩E=Ø. (N3) Define X2as the closure of the mean convex region of R3−(E∪D2). Then, M−Eis disjoint from X1∪X2. In particular, M−Eis contained in the halfspace {x3≤x3(D2)}. 51 To prove the claim and following the discussion in Sections 4.2 and 4.3, we will explain how to construct the disks D1, D2in each of the cases given by (G1) with M∞being a Riemann minimal example with horizontal limit tangent plane at infinity, or M∞being a vertical catenoid. In the first case, we simply take D1, D2as the horizontal disks bounded by almost-circles c0(ε), c1(ε)contained in the boundary of a piece R0(ε)⊂Eas defined in the paragraph just before Assertion 4.9. In the case (G1) with M∞being a vertical catenoid, we take D1, D2as the convex horizontal disks bounded by αnand βn, respectively (here we are using the notation in (12)). Properties (N1), (N2) hold from item 4 of Theorem 1.6. Concerning item (N3), if M−Eintersects X1then one can find a contradiction by adapting the arguments in paragraphs four and five of the proof of Theorem 1.3. Hence M−Eis disjoint from X1and similarly, M−Eis disjoint from X2. 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