Foliations on the open 3 -ball by complete surfaces
aa r X i v : . [ m a t h . G T ] D ec FOLIATIONS ON THE OPEN -BALL BY COMPLETESURFACES TAKASHI INABA AND KAZUO MASUDA
Abstract.
When is a manifold a leaf of a complete closed foliation on theopen unit ball? We give some answers to this question. Introduction and statement of results
Let F be a foliation on a Riemannian manifold ( M, g ). A leaf L of F is called closed if L is a closed subset of M (this is equivalent to say that L is properlyembedded ), and complete if L is complete with respect to the induced Riemannianmetric g | L . A foliation F is said to be closed (resp. complete ) if all leaves of F are closed (resp. complete). Let B n be the open unit ball of the Euclidean space R n with the induced Euclidean metric. Imitating Sondow [10], we consider thefollowing question: Given a connected smooth open manifold L , does there exist acomplete closed smooth foliation on B n with a leaf diffeomorphic to L ? Newnessof this question is to treat complete closed foliations on incomplete open manifolds.As one can imagine, in order to construct such foliations, one must “turbulize”(in a sense) all the leaves along the (ideal) boundary of B n . The motivation ofour work comes from recent papers of Alarc´on ([1], [2], [3]) about complete closedholomorphic foliations on the open ball of C n . We think that it is natural to treatthe same theme also in the real smooth category. Since real smooth objects aremuch more flexible than holomorphic ones, the problem becomes much easier todeal with in our setting. In this paper we will focus on the case of foliations of realcodimension one. Our first result is the following, whose holomorphic version hasbeen obtained by Alarc´on and Globevnik ([1], [3]). Theorem 1.1.
For any connected open orientable smooth surface Σ , there is acodimension complete closed smooth foliation on B with a leaf diffeomorphic to Σ . Remark.
In [6], Hector and Bouma showed the same statement on R . In [7],Hector and Peralta-Salas generalized it in higher dimensions. Remark.
The corresponding result to Theorem 1.1 for Sondow’s original question(i.e. the realization of manifolds as leaves of foliations on compact manifolds) wasfirst obtained by Cantwell and Conlon [5]. For recent developments in this area,see e.g. [4], [8].
Date : December 12, 2019.2010
Mathematics Subject Classification.
Key words and phrases. foliation, complete leaf, uni-leaf foliation. emark. A non-orientable surface cannot be a leaf of a foliation of B . In fact, ifit can, the foliation must be transversely non-orientable. The existence of such afoliation contradicts the simply-connectedness of B .In this paper we call a foliation F uni-leaf if all the leaves of F are mutuallydiffeomorphic. Our next result is concerned with uni-leaf foliations. For a connectedopen orientable surface Σ, let E be the set of ends of Σ with the usual topologyand E ∗ the closed subset of E consisting of nonplanar ends. It is known ([9]) thatthe pair ( E , E ∗ ) determines the diffeomorphism type of Σ. Here we introduce a newconcept. Let e be a point of E and let Z be a subset of E − E ∗ − { e } . Supposethat every point of Z is an isolated point of E and the derived set of Z in E is E ∗ . In this situation we say that the 4-tuple ( E , E ∗ , Z, e ) satisfies the self-similarityproperty if the following condition holds: there exist two copies ( E + , E + ∗ , Z + , e + ),( E − , E −∗ , Z − , e − ) of ( E , E ∗ , Z, e ) and homeomorphisms g ± : E → E ± , h : E + ∨ e + = e − E − → E such that g ± ( E ∗ ) = E ±∗ , g ± ( Z ) = Z ± , g ± ( e ) = e ± , h ( e ± ) = e and thatif e ∈ E ∗ then h ( E + ∗ ∨ e + = e − E −∗ ) = E ∗ , if e / ∈ E ∗ then h ( E + ∗ ⊔ E −∗ ) = E ∗ , and h ( Z + ⊔ Z − ) = Z , where ∨ is the wedge sum. (We notice here that Z is a countableset, because it consists of isolated points of the compact metrizable space E .)Now the second result of this paper is the following. Theorem 1.2.
Let Σ be a connected open orientable smooth surface and ( E , E ∗ ) itsendset pair. Suppose that there exist a point e of E and a subset Z of E − E ∗ − { e } such that (1) every point of Z is an isolated point of E , (2) the derived set of Z in E is E ∗ , and (3) ( E , E ∗ , Z, e ) satisfies the self-similarity property.Then, there exists a codimension complete closed smooth uni-leaf foliation of B having Σ as a leaf. Remark.
As we can easily observe, surfaces satisfying the property above abound.The following surface (Σ(2) in §
4) is an example. For more information on otherexamples, see §
4. By this theorem we have a lot of surfaces as leaves of uni-leaffoliations. ee e e e e e e e e e e e Figure 1. Σ(2) emark. A corresponding result with this theorem in the holomorphic situationhas been obtained by Alarc´on and Forstneriˆc [2], where the leaves are disks { z ∈ C | | z | < } . Whether a comparable result holds with leaves other than disks doesnot seem to be known at present.The authors would like to thank Ryoji Kasagawa and Atsushi Sato for theirinterest in our work and many helpful comments.2. Constructing complete foliations on the ball
The content of this section is a real smooth version of the argument developedin [2].Let { r k } and { s k } be sequences of real numbers satisfying 0 < s < r < s 1, and let B k (resp. S k ) be the closed ball (resp. the sphere) in R n centered at the origin with radius r k (resp. s k ). Put Γ k = S k − U ε ( p k ), where0 < ε ≪ s , p k = (0 , · · · , , ( − k s k ) and U ε ( p k ) is the open ε -neighborhood of p k in R n . A path γ : [0 , → B n is divergent if γ ( t ) leaves any compact set of B n as t → 1. Then, the following is evident. Lemma 2.1. Every divergent smooth path in B n avoiding S k ≧ k Γ k ( for some k ) has infinite length. Put P k = R n − × [ − k, k ]. Let Ω be an open subset of R n diffeomorphic to B n such that its image projected to the last coordinate of R n is unbounded. Then, wecan choose a sequence { C k } k ∈ N of subsets of Ω satisfying the following properties:(i) C k is diffeomorphic to the closed ball, (ii) C k ⊂ Int C k +1 , (iii) S ∞ k =1 C k = Ω.(iv) C k ⊂ P k +1 , (v) C k − P k = ∅ and Lemma 2.2. There exists a diffeomorphism Φ from Ω to B n such that for all k ∈ N (1 k ) Φ( C k ) = B k and (2 k ) Φ( P k ∩ C k ) ∩ Γ k = ∅ . Proof. We will enlarge the domain of definition inductively. First, define Φ on C so that it satisfies (1 ) and (2 ). This is possible because, by (v) above, C − P isnon-empty and Γ ⊂ Int B . Next, suppose Φ has already been defined on C k soas to satisfy (1 ℓ ) and (2 ℓ ) for ℓ ≦ k . Since Γ k +1 ⊂ B k +1 − B k and, by (v) above, C k +1 − P k +1 is non-empty, it is possible to extend the definition of Φ on C k +1 sothat Φ( C k +1 − P k +1 ) ⊃ Γ k +1 and that Φ( C k +1 − C k ) = B k +1 − B k . Then, we seethat the resulting Φ satisfies (1 k +1 ) and (2 k +1 ). Since { C k } k ∈ N is an exhaustingsequence of subsets of Ω, this inductive procedure gives a diffeomorphism from Ωto B n , as desired. ✷ Lemma 2.3. Let Φ be as in Lemma Then, for all k ∈ N , Φ( P k ∩ Ω) ∩ Γ k = ∅ . Proof. Since Γ k ⊂ B k = Φ( C k ), a point p of Ω satisfies Φ( p ) ∈ Γ k only if p ∈ C k .Hence, by Lemma 2.2 (2 k ), Φ( P k ∩ Ω) ∩ Γ k = Φ( P k ∩ C k ) ∩ Γ k = ∅ . ✷ Let G be a closed foliation on Ω (i.e. every leaf of G is a closed subset of Ω).Then, the pushout foliation F = Φ( G ) on B n is also a closed foliation. Here, weconsider the following property (P) for G :(P) For any leaf L of G there exists k ∈ N such that L ⊂ P k .We recall that a leaf F of F is complete if and only if every divergent smoothpath in F has infinite length. The next lemma gives a sufficient condition for thecompleteness of the leaves of F . emma 2.4. If G satisfies the property (P) , then all leaves of F are complete. Proof. Suppose G satisfies (P) and let F be any leaf of F . Put L = Φ − ( F ). Since L is a leaf of G , by (P) there exists k L ∈ N such that L ⊂ P k L . Then, noticingLemma 2.3, for any k ≧ k L we have F ∩ Γ k = Φ( L ) ∩ Γ k ⊂ Φ( P k L ∩ Ω) ∩ Γ k ⊂ Φ( P k ∩ Ω) ∩ Γ k = ∅ D Therefore, F does not intersect S k ≧ k L Γ k , hence, in particular,neither does any smooth path on F . This with Lemma 2.1 implies the completenessof the leaves of F . ✷ Realizing open surfaces as leaves Proof of Theorem 1.1. It is known that two smooth surfaces are diffeomorphicif and only if they are homeomorphic. And, according to [9, Theorem 1], thehomeomorphism type of a connected open orientable surface Σ is classified by itsgenus and the homeomorphism type of its endsets pair ( E , E ∗ ). (Recall that E is always compact and totally disconnected.) Thus, any such surface Σ can beconstructed as follows: First, remove from R a closed totally disconnected set X .( X ∪{∞} will be the endset E of Σ, where ∞ is the point of infinity of the one-pointcompactification of R .) Next, take a countable set Z in R − X in such a waythat for any compact set K in R − X the intersection Z ∩ K is finite. (The setof accumulation points of Z in R ∪ {∞} will be E ∗ .) Then, for each point q of Z ,choose a small compact metric neighborhood U q of q in R − X so that they arepairwise disjoint, and in each U q perform a surgery to make a genus. The resultingsurface is Σ. Observe that the whole of the above construction can be carried out in( R − X ) × R . To do so, for each q ∈ Z choose a small compact metric neighborhood V q of ( q, 0) in ( R − X ) × R , and perform ambient surgeries on ( R − X ) × { } insideeach V q . Thus, we obtain Σ as a properly embedded submanifold of ( R − X ) × R .Note that Σ separates ( R − X ) × R into two connected components.We then take a Morse function f : ( R − X ) × R → R so that(1) f ( x, y, z ) = z for ( x, y, z ) ∈ ( R − X ) × [( −∞ , − ∪ [1 , ∞ )], and that(2) 0 is a regular value of f with f − (0) = Σ.The existence of such f follows from the above construction of Σ. We let Crit( f )denote the set of critical points of f , (which is a countably infinite set if Σ hasnonplanar ends). Now, we take an increasing sequence ∅ = K ⊂ K ⊂ K ⊂ · · · of codimension 0 compact submanifolds in R − X such that S ∞ i =1 K i = R − X .For each p ∈ Crit( f ), we will construct an injective smooth path c p : [0 , ∞ ) → ( R − X ) × R as follows. Suppose p ∈ ( K i − K i − ) × R . Then,(1) c p (0) = p ,(2) c p intersects neither Σ nor ( R − X ) × {± } ,(3) c p does not intersect K i − × R ,(4) for each j ≧ i , c p intersects ∂K j × R transversely at exactly one point,(5) c p ( t ) converges to a point in ( X ∪ {∞} ) × {± / } as t → ∞ , and(6) if p = q , then c p ([0 , ∞ )) and c q ([0 , ∞ )) are disjoint.Such a choice is possible. Note that, by the conditions (3) and (4), the union [ p ∈ Crit( f ) c p ([0 , ∞ ))is a closed subset of ( R − X ) × R . Hence, the space M obtained from ( R − X ) × R by removing S p ∈ Crit( f ) c p ([0 , ∞ )) is an open submanifold of ( R − X ) × R . laim 1. M is diffeomorphic to ( R − X ) × R . Proof. Put N = D × [ − , ∞ ) (where D is the closed unit disk in R centered atthe origin). Take a nonnegative bounded smooth function λ : N → R satisfying(1) λ ( x, y, z ) = 0 if and only if x = y = 0 and z ∈ [0 , ∞ ), and(2) λ = 1 near ∂N .We define a smooth vector field V on N by V = λ ∂∂z and let ϕ : N × [0 , ∞ ) → N be the (local) flow generated by V . Then, we see that the map g : N → N − [ { } × [0 , ∞ )] defined as g ( x, y, z ) = ϕ (( x, y, − , z + 1) is a diffeomorphism which is theidentity near ∂N . Now, we can choose for each p ∈ Crit( f ) a smooth embedding u p : N → ( R − X ) × R so that(1) u p (0 , z ) = c p ( z ) for z ∈ [0 , ∞ ),(2) the diameter of u p ( D × { t } ) tends to 0 as t → ∞ ,(3) u p ( N ) intersects neither Σ nor ( R − X ) × {± } , and(4) if p = q , then u p ( N ) and u q ( N ) are disjoint.Then, we obtain a diffeomorphism h from ( R − X ) × R to M by setting h = u p ◦ g ◦ u p − on u p ( N ) for each p ∈ Crit( f ) and h = id otherwise. This proves theclaim. ✷ Now, we define an open subset Ω of R to be the union of M and R × (2 , ∞ ). Claim 2. Ω is diffeomorphic to B . Proof. Put Q = [( R − X ) × R ] ∪ [ R × (2 , ∞ )] and define a map k : Q → Ω as k = h on ( R − X ) × R and k = id on R × (2 , ∞ ), where h is the diffeomorphismgiven in the proof of Claim 1. Then, k is a diffeomorphism. Since B is obviouslydiffeomorphic to R , in order to prove the claim, we have only to show that Q isdiffeomorphic to R . Take a nonnegative bounded smooth function µ : R → R satisfying the condition: µ = 0 exactly on X × ( −∞ , W on R by W = µ ∂∂z and let ψ be the flow generated by W . Then, we see thatthe map ℓ : R → Q defined by ℓ ( x, y, z ) = ψ (( x, y, , z − 3) is a diffeomorphism.This proves the claim. ✷ Next, we extend the domain of our Morse function f to Ω by defining f to bethe projection to the second factor on R × (2 , ∞ ). We let G denote the foliationon Ω whose leaves are connected components of the level sets of f . Then, G hasno singularities because all the critical points of f are removed from Ω. It is alsoobvious that all leaves of G are closed in Ω. By the construction, we see that G satisfies the property (P) in § 2. Therefore, if we take a diffeomorphism Φ : Ω → B as in Lemma 2.2 and put F = Φ( G ), then F is a complete closed foliation on B containing Σ as a leaf. This completes the proof of Theorem 1.1. ✷ uni-leaf foliations In this section we consider the question: which manifold is a leaf of a completeclosed uni-leaf foliation on the open unit ball? This question has first been asked byAlarc´on and Forstneriˆc [2] in the holomorphic category. They have shown that forany integer n > 1, there exists a complete closed holomorphic uni-leaf foliation ofthe open unit ball in C n with disks as leaves. We work in the real smooth categoryand prove Theorem 1.2. roof of Theorem 1.2. Let Σ be a connected open orientable smooth surface and( E , E ∗ ) the endset pair of Σ. We use the notation in § 1. We assume that there exist e and Z as in Theorem 1.2 such that ( E , E ∗ , Z, e ) satisfies the self-similarity property.Put X = E − { e } , Y = E ∗ − { e } , X ± = E ± − { e ±} , and Y ± = E ±∗ − { e ±} . Via h ,we regard E ± , E ±∗ , X ± , Y ± and Z ± as subsets of E , E ∗ , X , Y and Z respectively.Now, we will start the construction of the uni-leaf foliation. First, we embed E into the one-point compactification R ∪ {∞} of R in such a way that e is mappedto ∞ . From now on, we identify e with ∞ and regard X , Y and Z as subsetsof R . Note that, by the properties of Z , X − Z is closed in R . Next, similarlyas in the previous section, for each point q of Z , choose a small compact metricneighborhood V q of ( q, 0) in the open 3-manifold ( R − ( X − Z )) × R so that theyare pairwise disjoint, and perform an ambient surgery on ( R − ( X − Z )) × { } to make a genus inside each V q . Then, ( R − ( X − Z )) × { } is modified and weobtain a new surface as a properly embedded submanifold of ( R − ( X − Z )) × R .Let ˇΣ denote this surface. We see that the endset pair of ˇΣ is ( E − Z, E ∗ ). We mayassume that for each q ∈ Z the intersection of ˇΣ and { q } × R is a single point.Here, let us recall the self-similarity property in Theorem 1.2. The spaces X , Y and Z are respectively expressed as the disjoint union of two subsets as follows: X = X + ⊔ X − , Y = Y + ⊔ Y − and Z = Z + ⊔ Z − , with the property that X , Y and Z are respectively homeomorphic to X ± , Y ± and Z ± .Now, we put O = R − ( X + − Z + ) × [ − , ∞ ) − ( X − − Z − ) × ( −∞ , . Let ν : R → [0 , / 2) be a smooth function such that(1) ν ( q ) = 0 if and only if q ∈ X − Z ,(2) ν ( q ) tends to 0 as k q k → ∞ and that(3) ν ( q ) = ν ( q ′ ) for any two different points q and q ′ of Z .We then take a Morse function f : O → R so that(1) f − (0) = ˇΣ,(2) Crit( f ) consists of the following points: ( q, − − ν ( q )) and ( q, ν ( q ) − ) foreach q ∈ Z + , ( q, − ν ( q ) − ) and ( q, ν ( q )) for each q ∈ Z − .(3) For each p ∈ Crit( f ), the value f ( p ) is the z -coordinate of p ,Let W + q ( q ∈ Z + ) be a small compact regular neighborhood of the segment { q } × [ − − ν ( q ) , ν ( q ) − ] in O , and W − q ( q ∈ Z − ) be a small compact regular neighborhoodof the segment { q } × [ − ν ( q ) − , ν ( q )] in O . We choose these neighborhoods soas to be mutually disjoint.(4) Inside each W + q ( q ∈ Z + ) or W − q ( q ∈ Z − ), f is conjugate to the standardMorse function which admits a standard canceling pair of critical points,the one which has a smaller z -coordinate is of index 1 and the other is ofindex 2.(5) The lines { q } × [ − − ν ( q ) , ∞ ) ( q ∈ Z + ) and { q } × ( −∞ , ν ( q )] ( q ∈ Z − )are transverse to the level sets of f everywhere except at critical points,(6) Outside the union of all W + q ’s ( q ∈ Z + ) and W − q ’s ( q ∈ Z − ), f is thestandard projection to the z -coordinate: f ( x, y, z ) = z .(7) The z -coordinate is bounded from above and below in each level set of f .Then, the family of the level sets of f define a singular foliation on O . The singu-larities are the critical points of f . By the choice of ν , we see that each level set ontains at most one critical point. We can also observe that for each z ∈ R theendset pair ( E z , E ∗ z ) of the level set f − ( z ) is identified with: ( E − Z, E ∗ ) × { z } if − ≦ z ≦ 1, ( E + − Z + , E + ∗ ) × { z } if z > 1, and ( E − − Z − , E −∗ ) × { z } if z < − E − Z, E ∗ ).As a final step, we defineΩ = O − [ {{ q } × [ − − ν ( q ) , ∞ ) | q ∈ Z + } − [ {{ q } × ( −∞ , ν ( q )] | q ∈ Z − } . Then, by the argument given in the proof of Theorem 1.1, we see that Ω is dif-feomorphic to R . Each level set L z = f − ( z ) ∩ Ω of f | Ω is obtained from f − ( z )by deleting the points of intersection with the segments S {{ q } × [ − − ν ( q ) , ∞ ) | q ∈ Z + } ∪ S {{ q } × ( −∞ , ν ( q )] | q ∈ Z − } . Since all the critical points of f areremoved by this deletion, every L z is now a non-singular smooth surface. Let G be the foliation on Ω thus obtained. Here, observe that if the point of intersectionof f − ( z ) and { q } × J ( q ∈ Z , and J is either [ − − ν ( q ) , ∞ ) or ( −∞ , ν ( q )])is not a critical point, then the deletion yields one puncture (or, one planar end)on f − ( z ), while if the point of intersection is a critical point, then the deletionyields two punctures (or, two planar ends). Now, let Z z be the set of all ends of L z newly produced by these deletions. Then, the endset pair of L z is expressedas ( E z ∪ Z z , E ∗ z ), where ( E z , E ∗ z ) is the endset pair of f − ( z ). Since, as remarkedabove, each f − ( z ) contains at most one critical point, it follows from the propertyof ν and the property (2) of f that Z z is identified with: Z if − ≦ z ≦ 1, theunion of Z + and F z if z > 1, the union of Z − and F z if z < − 1, where F z is a(possibly empty) finite subset of R − X . (Supplementary explanation: If z ≧ f − ( z ) does not intersect { q } × ( −∞ , ν ( q )] for any q ∈ Z − . So, in this case, F z is either a singleton or empty depending on whether there exists a critical pointon f − ( z ) ∩ { q } × [ − − ν ( q ) , ∞ ) for some q ∈ Z + . If 1 < z < 2, we see that f − ( z ) intersects { q } × ( −∞ , ν ( q )] for at most finitely many q ∈ Z − .) There-fore, ( E z ∪ Z z , E ∗ z , Z z , ∞ ), is identified with: ( E , E ∗ , Z, ∞ ) × { z } if − ≦ z ≦ E + ∪ F z , E + ∗ , Z + ∪ F z , ∞ ) × { z } if z > 1, and ( E − ∪ F z , E −∗ , Z − ∪ F z , ∞ ) × { z } if z < − Lemma 4.1. If F is a finite subset of R − X , then there is a homeomorphism h : E ∪ F → E such that h is the identity on E ∗ and that h ( Z ∪ F ) = Z . Proof. Let F be { x , · · · , x r } . Take a point p in E ∗ and a sequence { q n } ∞ n =1 in Z converging to p . We define a bijection h : E ∪ F → E by: h ( x k ) = q k for k = 1 , · · · , r , h ( q n ) = q r + n for n ≧ 1, and h is the identity otherwise. Then, h isobviously continuous at any point other than p , and the continuity of h at p is alsoobvious. This proves the lemma. ✷ By this lemma and the self-similarity property, the 4-tuple ( E z ∪ Z z , E ∗ z , Z z , ∞ )for the leaf L z is homeomorphic to ( E , E ∗ , Z, ∞ ) for every z ∈ R . Hence, we canconclude that all the leaves L z of Ω is diffeomorphic to Σ. Finally, if we follow ourprocedure described in § 2, we obtain a complete closed uni-leaf foliation having Σas a leaf. This completes the proof of Theorem 1.2. ✷ We easily observe that there are many surfaces whose endset pairs satisfy thehypothesis of Theorem 1.2. Here are some examples: In the case of planar surfaces, E ∗ is empty, hence, the set Z in the definition of the self-similarity property, mustalso necessarily be empty. Thus, to check the self-similarity, we have only to show hat E ∨ e E is homeomorphic to E for some e ∈ E . The following surfaces satisfysuch a property: R , R minus a discrete closed infinite set, R minus a Cantorset, and S minus a Cantor set. In the case of nonplanar surfaces, there are alsomany surfaces satisfying the self-similarity property. We give here the followingexample. For a positive integer r , let Σ( r ) be the connected open orientable surfacewhose endset pair ( E , E ∗ ) is described as follows (for Σ(2), see Fig. 1): E consistsof e, e i , e i i , · · · , e i i ··· i r , and E ∗ consists of e, e i , e i i , · · · , e i i ··· i r − , where eachsuffix i k (1 ≦ k ≦ r ) runs over all positive integers. For 1 ≦ ℓ ≦ r − 1, the ℓ -th derived set E ( ℓ ) of E consists of e, e i , e i i , · · · , e i i ··· i r − ℓ , and E ( r ) = { e } . e i converges to e as i → ∞ . For each 1 ≦ k ≦ r , e i i ··· i k converges to e i i ··· i k − as i k → ∞ while i , i , · · · , i k − being fixed.We show Σ( r ) satisfies the self-similarity property. Put X = E−{ e } , Y = E ∗ −{ e } and Z = E − E ∗ and set X + = { e i , e i i , · · · , e i i ··· i r | i is even and i , · · · , i r are arbitrary } ,X − = { e i , e i i , · · · , e i i ··· i r | i is odd and i , · · · , i r are arbitrary } ,Y + = { e i , e i i , · · · , e i i ··· i r − | i is even and i , · · · , i r − are arbitrary } ,Y − = { e i , e i i , · · · , e i i ··· i r − | i is odd and i , · · · , i r − are arbitrary } ,Z + = { e i i ··· i r | i is even and i , · · · , i r are arbitrary } ,Z − = { e i i ··· i r | i is odd and i , · · · , i r are arbitrary } , Define g + : X → X + by g + ( e i i ··· i k ) = e (2 i ) i ··· i k and g − : X → X − by g − ( e i i ··· i k ) = e (2 i − i ··· i k for 1 ≦ k ≦ r . This shows the self-similarity of Σ( r ). Question. List up all the open orientable surfaces whose endsets satisfy the self-similarity property. Question. Can a surface which does not satisfy the self-similarity property berealized as a leaf of a uni-leaf foliation on B ?As a final remark of this section, we show: Proposition 4.2. There are infinitely many codimension complete closed smoothuni-leaf foliations of the open unit ball B having R as a leaf such that the leafspaces of any two of them are not mutually homeomorphic. Proof. Let T be a tree (a connected graph without cycles) embedded in R suchthat the z -coordinate of T is unbounded, and that T satisfies the condition (C):no edge of T is horizontal (i.e. contained in R × { z } for some z ). Then, if we setΩ to be a regular neighborhood of T , we obtain a complete closed foliation F T on B with all leaves diffeomorphic to R . If two embedded trees T and T ′ are notisotopic through embedded trees satisfying (C), then leaf spaces of the foliations F T and F T ′ are not homeomorphic. Since there are infinitely many isotopy typesof such trees, the proof of the proposition is complete. ✷ Higher dimensional leaves In this section we consider the case of higher dimensional leaves. Let B n denotethe closed unit n -ball, and pr i the projection from a product space to its i -th factor. heorem 5.1. Let n ≧ . Suppose that F is a connected compact ( n − -dimensional smooth submanifold of B n − × R such that F ∩ ( ∂ B n − × R ) = ∂ B n − ×{ } = ∂F and that F is transverse to ∂ B n − × R at ∂F . Let E be a closed subsetof F satisfying that (1) F − E is connected, (2) E contains ∂F , and that (3) there exists a neighborhood U of E in F such that pr : B n − × R → B n − maps U diffeomorphically to pr ( U ) and that pr − pr ( U ) ∩ F = U .Then, there is a codimension complete closed smooth foliation of B n with a leafdiffeomorphic to F − E . Proof. The proof is essentially the same as the one in the surface case. Let n , F and E be as above. We take a Morse function f : ( B n − − pr ( E )) × R → R so that(1) f = pr on ( B n − − pr ( E )) × [( −∞ , − ∪ [1 , ∞ )], and that(2) 0 is a regular value of f with f − (0) = F − E .Next, we take an exhausting sequence { K i } of codimension 0 compact submanifoldsin B n − − pr ( E ), and a family of injective smooth paths c p : [0 , ∞ ) → ( B n − − pr ( E )) × R , p ∈ Crit( f ), satisfying the same six conditions with the ones in § 3. Then, M = ( B n − − pr ( E )) × R − S p ∈ Crit( f ) c p ([0 , ∞ )) is diffeomorphic to( B n − − pr ( E )) × R , and Ω = M ∪ ( B n − × (2 , ∞ )) is diffeomorphic to B n . Finally,by exactly the same argument given in § ✷ Remark. If E − ∂F is totally disconnected, then after a suitable isotopy on F , wecan always assume that E − ∂F is contained in a small collar neighborhood of ∂F .Then, the condition (3) in the Theorem is satisfied. Remark. F − E can be a pathological manifold. For example, we may take as E − ∂F the Whitehead continuum, the Menger sponge, and so on.6. Higher codimensions Proposition 6.1. Let q and q ′ be positive integers such that ≦ q < q ′ . Givena connected p -dimensional manifold L , if there is a codimension q complete closedsmooth foliation on B p + q with a leaf diffeomorphic to L , then, there is a codimension q ′ complete closed smooth foliation on B p + q ′ with a leaf diffeomorphic to L . Proof. Suppose F is a codimension q complete closed smooth foliation on B p + q with a leaf diffeomorphic to L . Then the foliation on B p + q × B q ′ − q defined by F × { z } ( F ∈ F , z ∈ B q ′ − q ) as leaves is a codimension q ′ complete closed smoothfoliation and has a leaf diffeomorphic to L . Since B p + q × B q ′ − q is diffeomorphic to B p + q ′ by a quasi-isometric diffeomorphism, the conclusion follows. ✷ Combining Proposition 6.1 with Theorem 1.1 and Theorem 5.1, we obtain Theorem 6.2. Let L be Σ in Theorem 1.1 or F − E in Theorem 5.1, and let p = dim L . Then, for any positive integer q , there is a codimension q completeclosed smooth foliations on the open unit ball B p + q having L as a leaf. Similarly, by Proposition 6.1 and Theorem 1.2 we have heorem 6.3. Let L be Σ in Theorem 1.2. Then, for any positive integer q , thereis a codimension q complete closed smooth uni-leaf foliation on the open unit ball B q having L as a leaf. References [1] A. Alarc´on, Complete complex hypersurfaces in the ball come in foliations ,arXiv:1802.02004, 2018.[2] A. Alarc´on and F. Forstneriˆc, A foliation of the ball by complete holomorphic discs ,arXiv:1905.09878, 2019.[3] A. Alarc´on and J. Globevnik, Complete embedded complex curves in the ball of C canhave any topology , Anal. PDE, (2017), 1987–1999.[4] J. A. ´Alvarez L´opez and R. Barral Lij´o, Bounded geometry and leaves , Math. Nachr. (2017), 1448–1469.[5] J. Cantwell and L. 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