On the topology of surfaces with the generalised simple lift property

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia (J Differ Geom 102(1):1–23, 2016) and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of property embedded minimal disks satisfy the generalised simple lift property.


Introduction
Motivated by the work of Colding and Minicozzi [3][4][5][6] and Hoffman and White [8] on minimal laminations obtained as limits of sequences of properly embedded minimal disks, in [1] Bernstein and Tinaglia introduce the concept of the simple lift property. Interest in these surfaces arises because leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the simple lift property. In [1] they prove that an embedded minimal surface ⊂ with the simple lift property must have genus zero, if is an orientable three-manifold satisfying certain geometric conditions. In particular, one key condition is that cannot contain closed minimal surfaces.
In this paper, we generalise this result by taking an arbitrary orientable three-manifold and introducing the concept of the generalised simple lift property, which extends the simple lift property in [1]. Indeed, we prove that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised simple lift property and we are able to restrict the topology of an arbitrary surface ⊂ with the generalised simple lift property.
Among other things, we prove that the only possible compact leaves of a minimal lamination obtained as limits of a sequence of properly embedded minimal disks are the torus in the orientable case, the Klein bottle and the connected sum of three and four projective planes in the non-orientable case.

Notation and definitions
Throughout the paper, we will assume to be an open subset of an orientable threedimensional Riemannian manifold (M, g). We denote by dist the distance function on and by exp the exponential map. Therefore, we have where B r (0) is the Euclidean ball in R 3 of radius r centred at the origin, and B r ( p) is the geodesic ball in M of radius r centred at p ∈ .
For an embedded surface , we write exp ⊥ : N → to denote the norma exponential map, where N is the normal bundle. If N is trivial, then we say that is two-sided, otherwise we say that is one-sided. As is oriented, being two-sided is equivalent to saying that is orientable.
Let us fix a subset U ⊂ N , then we define where v ⊥ is orthogonal to v T , and v T is perpendicular to the fibres of . We say that such v is δ-parallel to if |v ⊥ | ≤ δ|v| and Given > 0, we set U := {( p, v) ∈ N | |v| < } and define N ( ), theneighbourhood of , to be N U ( ). If is an embedded smooth surface and 0 ⊂ is a pre-compact subset, then ∃ > 0 so that N ( 0 ) is regular.
Given a fixed embedded surface and δ ≥ 0, we say that another embedded smooth surface is a smooth δ-graph over if there exists an > 0 such that: 1. N ( ) is a regular -neighbourhood of ; 2. either is a proper subset of N ( ) or is a proper subset of N ( ) \ ; 3. for all (q, v) ∈ T is δ-parallel to .
We say that a smooth δ-graph over is a smooth δ-cover of , if is connected and ( ) = .
Let γ : [0, 1] → be a smooth curve in . We will also denote the image of such γ as γ .
We say that a curve γ : This definition extends to piece-wise C 1 curves in an obvious manner.

The generalised simple lift property for a finite number of curves
Let us introduce the concept of lifts of curves into embedded disks.

Definition 3.1 Generalised simple lift property.
Let be a surface in . Then has the generalised simple lift property if, for any δ > 0 and for any p ∈ , the following holds.
One should notice that the embedded disk ⊂ that the definition implies exists will depend on the choice of the constant δ > 0, the n curves γ 1 , . . . , γ n and the pre-compact subset U ⊂ that contains the curves. Notation wise, throughout this paper, when studying a lift of n given curves γ 1 , . . . , γ n , if we want to highlight the dependence of the construction on the choice of curves, we will denote the embedded disk that contains the generalised simple δ-lift of γ 1 ∪ · · · ∪ γ n by (γ 1 , . . . , γ n ).
A surface with the generalised simple lift property is one for which, in an effective sense, the universal cover of the surface can be properly embedded as a disk near the surface. For this reason, to understand the topology of the surface , it is important to understand the lifting behaviour of closed curves.
With this in mind, we give the following definition. If a closed curve γ has the closed lift property, then there is a sequence δ i → 0 so that there are closed simple δ i -lifts γ i of γ .
If it is possible to choose the lifts of a curve γ to be embedded (and in particular nonintersecting), we say γ has the embedded (closed/open) lift property.

Remark 3.3
In Proposition 3.4 below and in Lemma 4.2 we will be constructing the lift of two (or more) simple closed curves intersecting at one point by considering the union of these curves as a single curve.
The proposition below, which we will call Lifting Lemma, is analogous to Proposition 4.4 in Bernstein and Tinaglia's paper [1].

Proposition 3.4 Lifting lemma
Let ⊂ be an embedded surface with the generalised simple lift property. Let us take into consideration two closed, smooth curves Proof Let us take into consideration the curve μ = α • β • α −1 • β −1 as defined above.
By assumption, we can consider the embedded disk (α, β) which contains opens lifts of both α and β.
By re-parametrising appropriate restrictions of μ , we can write where the α , β , α −1 , β −1 : [0, 1] → are the δ-lifts of α, β, α −1 and β −1 respectively. Let us now pick a small simply-connected neighbourhood V of the point p = μ(0) such that V ⊂ U . By construction, (α, β) is an embedded disk, which means that we can order by height the components of −1 (V ) ∩ (α, β), where is the usual projection map onto . We will denote these ordered components as The number n of components will of course depend on the choice of δ > 0 and (α, β).
By construction, we then have: Without loss of generality, one can assume μ to be the generalised simple δ-lift of μ pointed at ( p, q) with q = α(0) . Moreover, a priori, these points will all belong to different components of −1 (V ) ∩ (α, β) and we will denote them as: where l is a function of j over the natural numbers, that is l = l( j) ∈ N.
Using this function l, we will study the signed number of sheets between the end points of the lifts of the curves α, α −1 , β and β −1 : By assumption, both α and β are open lifts, so that We will now prove that , and therefore that μ is closed.
Let us consider the two following cases separately: Without loss of generality, we can assume in this case that both numbers are positive: m[α], m[β] > 0. Then, using the fact that the disk (α, β) is embedded and that U is twosided, one can consider a disjoint family of parallel lifts of α, which we will denote by α [i].
The first member of this family is α [0] = α and the subsequent representatives of the family are those lifts α [i] of α such that α [i](0) will belong to V l(0) + i , which is the lift that starts i sheets above α(0) = q. By the embeddedness of (α, β) and the two-sidedness of U , the signed number of graphs between α [0](t) and α [i](t) is constant in t, so that also the lifts α [i] also have endpoints i sheets above the endpoint of α . We will now proceed to study the topology of surfaces with the generalised simple lift property.

The topology of embedded surfaces with the generalised simple lift property
The geometrical example at the centre of this initial topological study is the double torus minus a disk, that is the connected sum of two tori with a disk removed (see Fig. 1). By the classification of compact surfaces, we know that compact orientable surfaces are either the sphere S 2 or the connected sum of n tori, T 2 # · · · #T 2 , while non-orientable surfaces are given by the connected sum if n projective planes RP 2 # · · · #RP 2 . This classification extends to non-compact surfaces by taking into consideration boundary components.

Remark 4.1
In order to simplify the notation, we will denote by T 2 n the connected sum of n tori, and by RP 2 n the connected sum of n projective planes.
Arguing by contradiction, let us assume that both curves admit closed δ-lifts on an embedded disk (γ 1 , γ 2 ). In other words, there exists a choice of δ > 0 and U ⊂ pre-compact open subset containing both γ 1 and γ 2 , such that the embedded disk = (γ 1 , γ 2 ) given by Definition 3.1 contains γ 1 and γ 2 two closed δ-lifts of γ 1 and γ 2 respectively.
This represents a contradiction to the mod 2 degree theorem applied to the Jordan-Brouwer separation theorem. This contradiction finishes the proof of the lemma.
In the following claims, the surface ⊂ that we are considering is homeomorphic to T 2 #T 2 \ D and γ 1 : [0, 1] → denotes the smooth, non-separating Jordan curve in Fig. 2 . We will prove that a surface with the generalised simple lift property cannot contain an open subset homeomorphic to a double torus minus a disk by proving that γ 1 cannot have either a closed or an open lift in an embedded disk for a specific choice of five non-separating Jordan curves γ 1 , γ 2 , γ 3 , γ 4 , γ 5 (see Fig. 3). Proof Given the five smooth Jordan curves γ 1 , γ 2 , γ 3 , γ 4 , γ 5 : [0, 1] → pictured in Fig. 3, for an arbitrary δ > 0 and for an arbitrary pre-compact open set U ⊂ that contains these five curves γ i , we are considering the embedded disk = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ) for which the connected component of ∩ N (U ) that contains the δ-lifts γ i is a δ-cover of U (see Definition 3.1).
If either γ 2 •γ 3 or γ 3 •γ −1 2 has an embedded closed lift on , then we reach a contradiction by applying the same reasoning used in Lemma 4.2, since γ 1 (which we are assuming has a closed lift on ) and the given curve intersect transversally in a single point: admits an embedded closed lift, then one can find three values t 1 , t 2 , t 3 ∈ (0, 1) for which p = γ 1 (t 1 ) = γ 2 (t 2 ) = γ −1 2 (t 3 ). Following the construction of the lifting lemma, we consider the two-sided subset U ⊂ which in particular contains γ 1 , γ 2 and γ 3 , and pick a small simply-connected neighbourhood V of p contained in U , so that we can construct a family of parallel components of the lifts of V that can be ordered by height: −1 (V ) ∩ = {V (1), . . . , V (n)}. We can now consider the generalised simple δ-lift γ 1 The fact that p is the only point of intersection results from the following remark. γ 1 is indeed a one-cover of γ 1 , while on the other hand γ −1 2 (t 3 ) ∈ α belongs to a components of −1 (V ) ∩ that is different to that of p = γ 1 (t 1 ) = γ 2 (t 2 ) . In fact, if we denote by V (l 1 ) the component of −1 (V ) ∩ that contains p , we have that the component that contains γ −1 2 (t 3 ) will have height l 2 given by: since γ 3 is an open lift on (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ). Therefore, we constructed two closed curves γ 1 and α which intersect transversally on the disk = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ) in a single point p , which represents a contradiction to the mod 2 degree theorem applied to the Jordan Brouwer separation theorem. Therefore, the γ 3 must have a closed δ-lift on = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ).
Let us now take into consideration the loop γ 4 which intersects γ 2 transversally in one single point. Arguing like before, we obtain that γ 4 will have a closed δ-lift on = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ) too. We have then constructed two smooth non-separating Jordan curves γ 3 and γ 4 that intersect transversally in one single point and both have a closed δ-lift on the disk . By Lemma 4.2, this represents a contradiction to the Jordan Brouwer separation theorem.
This implies that the initial curve γ 1 cannot have a closed δ-lift on the embedded disk = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ). Proof Just like in the previous claim, we are working with the same five curves in Fig. 3, and for any arbitrary δ > 0 and for an arbitrary pre-compact open set U ⊂ that contains these curves γ i we are considering the embedded disk = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ) given by Definition 3.1, for which the connected component of ∩ N (U ) that contains the δ-lifts γ i is a δ-cover of U . Arguing by contradiction like before, we will now assume that γ 1 admits an open δ-lift on such a disk = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ).
Let us consider the curve γ 2 : [0, 1] → which intersects γ 1 transversally in a single point. We will be considering the two cases of γ 2 admitting either an open or a closed δ-lift on = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ), and we will prove that both of them yield a contradiction.
If γ 5 admitted an open lift, the curves γ 3 and γ 5 would satisfy the hypotheses of the lifting lemma, and we could then apply the same argument as in the previous claim to the two curves γ 2 and γ 3 • γ 5 • γ −1 3 • γ −1 5 , both of which have a closed lift on and intersect transversally at a single point, hence obtaining a contradiction.
If instead γ 5 admitted a closed lift, then the two curves γ 3 and γ 4 would satisfy the hypotheses of the lifting lemma, and we could apply still the same argument as in Claim 4.3 to the two curves γ 5  has an open lift on . This simply follows from the fact that we would be able to apply the lifting lemma to the two curves γ 2 and γ 3 . So if γ 5 had a closed lift on , we would obtain two curves γ 2 • γ 3 • γ −1 2 • γ −1 3 and γ 5 intersecting transversally at one single point, both admitting a closed δ-lift on , which is a contradiction as already shown in the previous claim.
By applying the lifting lemma to the pairs of curves {γ 1 , γ 2 } and {γ 3 , γ 5 } respectively, we obtain two curves with a closed lift on = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ), namely α : . As already pointed out in the previous claim, considering the first couple of curves {γ 1 , γ 2 }, one of the curves α, γ 1 • γ 2 and γ 2 • γ −1 1 has an embedded closed lift, and likewise for the other couple {γ 3 , γ 5 }. In this proof, we will take into consideration only the most complicated case where α and β are the loops with the embedded closed lift on . One should argue just like in Claim 4.3 for the other cases.
Let us take into consideration the point of intersection { p} = α ∩ β. One should notice that there exist four values t 1 , t 2 , t 3 , t 4 ∈ (0, 1) such that . Let us now take as a two-sided pre-compact open set U ⊂ that contains both curves α and β, the original subset U ⊂ used to construct = (γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ). Following the construction in the lifting lemma, we can pick a small simply-connected neighbourhood V ⊂ U of p, so that we can produce a family of parallel components on of the lifts of V that can be ordered by height: All the curves γ i are the open lifts, so that -still following the notation of the lifting lemma m[γ j ] = 0 ∀ j = 1, . . . , 4 , which means that there are two cases: either m[ In the first case, we will consider the generalised simple lift α ∪ β based at (γ 2 (t 1 ) = γ 3 (t 3 ), p ) on the disk , which means there exists at least a point p ∈ N ( p) ∩ such that p ∈ γ 2 ∩ γ 3 .
We are left to prove that this point p is the only point of intersection between α and β . By construction, γ 2 (t 1 ) and γ 3 (t 3 ) belong to the same component of −1 (V ) ∩ , namely V (l 1 ). The other two points γ −1 2 (t 2 ) and γ −1 3 (t 4 ) will then belong to two other components V (l 2 ) and V (l 3 ) respectively. Moreover, since the number of components between α [k](0) and α [k](t) does not depend on k, we have that the heights of these two components will be: Hence 5 ] > 0, which means that p is indeed the only point of intersection between the two closed lifts α and β , which is a contradiction.
From these claims, we obtain the following result. By the classification of compact surfaces, we have that orientable surfaces are homeomorphic to S 2 or the connected sum of n tori, T 2 n , while non-orientable compact surfaces are homeomorphic to the connected sum of n projective planes, RP 2 n . Moreover, one should notice that in the non-orientable case we have the following homeomorphisms:

Minimal laminations
Let us now apply the results of the previous section to the case of minimal laminations. Let us first recall some facts about laminations. Definition 5.1 A subset L ⊂ is a smooth lamination if for each p ∈ L, there is a radius r p > 0, maps φ p , ψ p : B r p ( p) → B 1 (0) ⊂ R 3 and a closed set T p ⊂ (−1, 1) with 0 ∈ T p such that: (D 1 (0)) . We refer to maps φ p satisfying properties 1) and 2) as smoothing maps of L and to maps ψ p satisfying properties 1) and 3) as straightening maps of L.
A smooth lamination L ⊂ is proper in if it is closed, that is L = L. Any embedded smooth surface is a smooth lamination that is proper if and only if the surface is proper.
Definition 5.2 Let L ⊂ be a non-empty smooth lamination. A subset L ⊂ L is a leaf of L if L is a connected, embedded surface and for any p ∈ L , ∃ r p > 0 and a smoothing map φ p so that D 1 = φ p (L ∩ B r p ( p)) . For each p ∈ L, we will denote by L p the unique leaf of L containing p.
A smooth lamination L is a minimal lamination if each one of its leaves is minimal.
The following is the natural compactness result for sequences of properly embedded minimal surfaces with uniformly bounded second fundamental form (see for instance Appendix B in [6] for a proof).

Theorem 5.3 Let { i } i∈N be a sequence of smooth minimal surfaces, properly embedded in
. If for each compact subset U ⊂ there is a constant C(U ) < ∞ so that then, ∀ α ∈ (0, 1), up to passing to a subsequence, the i s converge in C ∞,α loc ( ) to L, a smooth proper minimal lamination in . Clearly, reg(S) is an open subset of , while sing(S) is closed in . In general, sing(S) ⊂ \reg(S) is a strict inclusion, however, by Lemma I.1.4 in [6] there exists a subsequence S of S so that = reg(S ) ∪ sing(S ). Without loss of generality, we will then consider sequences S that admit this decomposition. This work will be centred around limit laminations of minimal disk sequences, so it will be convenient to introduce the following definition (inspired by [17]).

Definition 5.6
Let us take a closed set K ⊂ in our ambient Riemannian three-manifold . Let us introduce a smooth proper minimal lamination L in \ K and a sequence S = { i } i∈N of properly embedded minimal disks in .
We will refer to the quadruple ( , K , L, S) as a minimal disk sequence if i. sing(S) = K , and ii. i \ K converge to L in C ∞,α loc ( \ K ), for some α ∈ (0, 1). The case where the i are assumed to be disks has been extensively studied and some structural results have been proved on the possible singular sets K and limit laminations L of a minimal disk sequence ( , K , L, S). For example, in [3][4][5][6] Colding and Minicozzi show that K must be contained in a Lipschitz curve and that for any point p ∈ K there exists a leaf of L that extends smoothly across p. When = R 3 , they further show that either K = ∅ or L is a foliation of R 3 \ K by parallel planes and that K consists of a connected Lipschitz curve which meets the leaves of L transversely. Using this result, Meeks and Rosenberg showed in [14] that the helicoid is the unique non-flat properly embedded minimal disk in R 3 . This uniqueness was then used by Meeks in [13] to prove that if = R 3 and K = ∅, then K is a line orthogonal to the leaves of L, which is precisely the limit of a sequence of rescalings of a helicoid.
For an arbitrary Riemannian three-manifold, such a simple description is not possible. In [2], Colding and Minicozzi construct a sequence of properly embedded minimal disks in the unit ball B 1 (0) ⊂ R 3 which has K = {0} and whose limit lamination consists of three leaves: two non-proper disks that spiral into the third, which is the punctured unit disk in the x 3 -plane. Inspired by this example, more cases have been constructed where the singular set K consists of any closed subset of a line [7,9,11,12], as well as examples where K is curved [15]. Finally, Hoffman and White [10] have also constructed minimal disk sequences in which K = ∅ and the limit lamination L has a leaf which is a proper annulus in . Proof Given L a leaf of L, if L is a disk, the curves γ i in L are themselves their own simple δ-lifts in any pre-compact open set U ⊂ L that contains them. Hence the proposition holds trivially, with q = p.
In the more general case, when L is not a disk, it is sufficient to prove the existence of a generalised simple lift of a single curve γ . By Proposition B.1 in Appendix B of [6], we obtain a bound on the Lipschitz norms of the straightening maps, which implies that for each pre-compact open subset U ⊂ L, there is a constant C = C(U ) such that Cλ ∈ (0, 1), and then for each i ∈ S, N λ (U ) ∩ i is a (possibly empty) Cλ-graph over U . Given a curve γ : [0, 1] → L contained in an open pre-compact subset U ⊂ L, let us denote by l the length of γ and d the diameter of U . For any δ > 0, choose > 0 such that C < min{1, δ}. Let μ = 3 4 exp(−2C(l + d)) and pick μ ∈ S such that N μ ( p) ∩ μ = ∅, where p = γ (0). Let be a component of μ ∩ N (U ) which contains a point q ∈ N μ ( p) ∩ . We have chosen > 0 so that μ ∩ N (U ) is a δ-graph over U . We claim that is a δ-cover of U containing a δ-lift of γ . This follows by showing that any curve in U of length at most 2(l + d) starting at p has a lift in starting at q. By construction, this lift is necessarily a δ-lift.
This result then implies:

Remark 5.9
By applying a lifting argument, one can further rule out the sphere S 2 and the projective plane RP 2 .