A Minimal Lamination with Cantor Set-Like Singularities

Given a compact closed subset $M$ of a line segment in $\mathbb{R}^3$, we construct a sequence of minimal surfaces $\Sigma_k$ embedded in a neighborhood $C$ of the line segment that converge smoothly to a limit lamination of $C$ away from $M$. Moreover, the curvature of this sequence blows up precisely on $M$, and the limit lamination has non-removable singularities precisely on the boundary of $M$.


introduction
Let Σ k ⊂ B R k = B R k (0) ⊂ R 3 be a sequence of compact embedded minimal surfaces with ∂Σ k ⊂ ∂B R k and curvature blowing up at the origin. In [1], Colding and Minicozzi showed that when R k → ∞, a subsequence converges off a Lipshitz curve to a foliation by parallel planes. In particular, the limit is a smooth, proper foliation. By contrast, in [2] Colding and Minicozzi constructed a sequence as above with R k uniformly bounded and converging to a limit lamination of the unit ball with a non-removable singularity at the origin. Later, B. Dean in [3] found a similar example where the limit lamination has a finite set of singularities along a line segment, and S. Khan in [4] found a limit lamination consisting of a non-properly embedded minimal disk in the upper half ball spiraling into a foliation by parallel planes of the lower half ball. Both Dean and Khan used methods that are analogous to those in [1]. Recently, using a variational method, D. Hoffman and B. White in [5] were able to construct a sequence converging to a non-proper limit lamination and with curvature blowup occurring along an arbitrary compact subset of a line segment. In this paper we do the same, but with a method that is derivative of that in [1] and [4]. The main theorem is: Theorem 1. Let M be a compact subset of {x 1 = x 2 = 0, |x 3 | ≤ 1/2} and let C = {x 2 1 +x 2 2 ≤ 1, |x 3 | ≤ 1/2}. Then there is a sequence of properly embedded minimal disks Σ k ⊂ C with ∂Σ k ⊂ ∂C and containing the vertical segment {(0, 0, t)||t| ≤ 1/2} so that: Moreover, Σ I \ {x 3 − axis} = Σ 1,I ∪ Σ 2,I , for ∞-valued graphs Σ 1,I and Σ 2,I each of which spirals in to the planes {x 3 = t 1 } from above and {x 3 = t 2 } from below.
It follows from (D) that a subsequence of the Σ k \ M converge to a limit lamination of C \ M. The leaves of this lamination are given by the multi-valued graphs Σ I given in (D), indexed by intervals I of the complement of M, taken together with the planes {x 3 = t} ∩ C for (0, 0, t) ∈ M. This lamination does not extend to a lamination of C, however, as every 1 boundary point of M is a non-removable singularity. Theorem 1 is inspired by the result of Hoffman and White in [5]. The author would like to thank Professor William P. Minicozzi who suggested to him the problem of constructing an example with singularities on the Cantor set (a special case of Theorem 1) Throughout we will use coordinates (x 1 , x 2 , x 3 ) for vectors in R 3 , and z = x + iy on C. For p ∈ R 3 , and s > 0, the ball in R 3 is B s (p). We denote the sectional curvature of a smooth surface Σ by K Σ . When Σ is immersed in R 3 , A Σ will be its second fundamental form. In particular, for Σ minimal we have that |A Σ | 2 = −2K Σ . Also, we will identify the set M ⊂ {x 3 -axis} with the corresponding subset of R ⊂ C; that is, the notation will not reflect the distinction, but will be clear from context. Our example will rely heavily on the Weierstrass Representation, which we introduce here.

The Weierstrass representation
Given a domain Ω ⊂ C, a meromorphic function g on Ω and a holomorphic one-form φ on Ω, one obtains a (branched) conformal minimal immersion F : Ω → R 3 , given by (c.f. [6]) the so-called Weierstrass representation associated to Ω, g, φ. The triple (Ω, g, φ) is referred to as the Weierstrass data of the immersion F . Here, γ z 0 ,z is a path in Ω connecting z 0 and z. By requiring that the domain Ω be simply connected, and that g be a non-vanishing holomorphic function, we can ensure that F (z) does not depend on the choice of path from z 0 to z, and that dF = 0. Changing the base point z 0 has the effect of translating the immersion by a fixed vector in R 3 . The unit normal n and the Gauss curvature K of the resulting surface are then (see sections 8, 9 in [6]) n = 2Re g, 2Im g, |g| 2 − 1 / |g| 2 + 1 , Since the pullback F * (dx 3 ) is Re φ, φ is usually called the height differential. The two standard examples are giving a catenoid, and giving a helicoid. We will always write our non-vanishing holomorphic function g in the form g = e ih , for a potentially vanishing holomorphic function h, and we will always take φ = dz. For such Weierstrass data, the differential dF may be expressed as 3. An outline of the proof Fix a compact subset M of the real line. We will be dealing with a family of immersions F k,a : Ω k,a → R 3 that depend on a parameter 0 < a < 1/2 given by Weierstrass data of the form Ω k,a , G k,a = e iH k,a , φ = dz, and a sequence M k ⊂ M that converge to a dense subset of M. Each function H k,a will be real valued when restricted to the real line in C. That is, writing H k,a = U k,a + iV k,a for real valued functions U k,a , V k,a : Ω k → R, we have that H k,a (x, 0) = U k,a (x, 0). Moreover, we will show that V k,a (x, y) > 0 when y > 0. A look at the expression for the unit normal given above in (2) then shows that all of the surfaces Σ k,a := F k,a (Ω k,a ) will be multi-valued graphs over the (x 1 , x 2 ) plane away from the x 3 -axis (since |g(x, y)| = 1 is equivalent to y = 0). The dependence on the parameter 0 < a < 1/2 will be such that lim a→0 |A Σ k,a | 2 (p) = ∞ for all p ∈ M k , and such that |A Σ k,a | 2 remains uniformly bounded in k and a away from M. We will then choose a suitable sequence a k → 0, and set F k = F k,a k , Ω k = Ω k,a k , G k = G k,a k , and H k = H k,a k . Immediately, (A), (B) and (C) of Theorem 1 are satisfied by the diagonal subsequence. In fact, we will show that any suitable sequence is a sequence a k → 0 satisfying a k < γ −k for a parameter γ > 1 which we introduce later. The bulk of the work will go towards establishing (D). To this end, we will show that This gives that the immersions F k : Ω k → R 3 are actually embeddings, and that the surfaces Σ k given by F k (Ω k ) are all embedded in a fixed cylinder This will then imply that the surfaces Σ k converge smoothly on compact subsets of C r 0 \M to a limit lamination of C r 0 . The claimed structure of the limit lamination(that is, that on each interval of the complement it consists of two multi-valued graphs that spiral into planes from above and below) will be established at the end. Note that this induces an identification of the closed set M, thought of as lying in the complex plane along the real axis, with its image in the x 3 -axis.
Throughout the paper, all computations will be carried out and recorded only on the upper half plane in C, as the corresponding computations on the lower half plane are completely analogous. By scaling it suffices to prove Theorem 1 (D) for some C r 0 , not C 1 in particular.

definitions
Let M ⊂ [0, 1] be a closed set. Fix γ > 1, and take M −1 to be the empty set. Then for k a non-negative integer, we inductively define two families of sets m k , and M k as follows: Assuming M k−1 is already defined, take m k to be any maximal subset of M with the property that, for p, q ∈ m k , r ∈ M k−1 , it holds that |p−q|, |p−r| ≥ γ −k . Then define M k = M k−1 ∪m k and M ∞ = ∪ k M k . Also, for x ∈ R define p k (x) to be the closest element in M k to x. Note that there are at most two such points, and we take p k (x) to be the closest point on the left, equivalently the smaller of the two points. For p ∈ M ∞ , we define e(p) to be the unique natural number such that p ∈ m e(p) . Note that e(p k (x)) ≤ k. We take and y 0,a (x) = ǫ x 2 + a 2 5/4 for ǫ to be determined. For p ∈ R we define h p,a (z) = h a (z − p) = u p,a (z) + iv p,a (z) and y p,a (x) = y 0,a (x − p).
We then take and y l,a (x) = min p∈m l y p,a (x).
We take for a parameter µ > γ to be determined. We take Y k (x) = min l≤k y l,a k (x).
We take and lastly set Ω ∞ = ∩ k Ω k . Note that in the above definitions, objects bearing the subscript "k" (as opposed to "l") always enumerate an (as yet undetermined) diagonal sequence. Consequently, the dependence on the parameter a is omitted from the notation. At times, the dependence on a will be suppressed from the notation for objects without the subscript "k". Also, note that for each x we have that Y k (x) = y p k ,a k (x). Again, when it is clear, the subscript "a k " will be suppressed. Keep in mind throughout that {a k } will always denote a sequence with a k ≤ γ −k . Also, the parameters γ and µ introduced in this section must satisfy µ 2/3 < γ < µ < γ 3 . The reasons are technical, and should become clear later in the paper.

Preliminary Results
We record some elementary properties of the sets M k and m k defined above which will be needed later.
Proof. Let p 1 < . . . < p n be n distinct elements of m k , ordered least to greatest. By construction we have that p k+1 − p k ≥ γ −k . Also, since p 1 , p n ∈ M we get Then there is a q ∈ M, and a positive integer k such that |p − q| > γ −k , ∀p ∈ M ∞ . In particular, this implies that m k is not maximal.
In order to avoid disrupting the narrative, the proofs of the remaining results in this section will be recorded later in the Appendix at the end. The proofs are somewhat tedious, though easily verified.

Lemma 6. .
For ǫ sufficiently small, h p (z) is holomorphic on ω p , h l is holomorphic on ω l , and H k is holomorphic on Ω k .
We will also need the following estimates: 6 Lemma 7. On the domain ω p it holds that: Integrating the above estimates from 0 to the upper boundary of ω p gives These estimates immediately give Corollary 8. We have the bounds where q k (x) is defined by the last equality above.

Proof of Lemma 2
We will first concern ourselves with establishing Lemma 2. (a) follows from (1) and the choice of z 0 = 0. Choosing ǫ < ǫ 0 < c ′−1/2 1 , where c ′ 1 is the constant in (11), and using (7) we get Here we have used that cos(1) > 1/2. This gives that all of the maps F k : Ω k → R 3 are indeed embeddings (for all values of a) and proves (b) of Lemma 2. Now, integrating (13) from Y k (x)/2 to Y k (x) gives Using the bound for V k recorded in (12), we get that k e 1 2 ǫc 2 q k (x) . We will show that r k (x) remains uniformly large in k; this establishes (c) of Lemma 2. First, we need Lemmas (9) and (10) below. In the following, take Φ(ξ) = ξ 5/3 e 1 2 2 c 2 ǫξ −1 .
Proof. The assumptions immediately give that Applying (16) and using that e(p k (x)) ≤ k we find that Equivalently, since we have chosen ασ − 5/3 ≥ 1, and we may assume δ < 1.
We are ready to prove: Lemma 11. (Lemma 2 (c)) There exists a sequence {c k } with c k > 0 and ∞ l=0 c l > 0 such that if r k (x) < 1, then r k (x) > c k r k−1 (x).
The immediate corollary is This establishes (c) of Lemma 2

Proof of Theorem 1(A), (B) and (C)
Note that (3) and our choice of Weierstrass data gives that For p ∈ m l , it is clear that F k (p) = (0, 0, p) for all k. Thus, for k > l we can then estimate since V k (x, 0) = 0 for all x ∈ R and hence |A Σ k (p)| 2 → ∞. For x ∈ M \ M ∞ , consider the sequence of points p l (x) ∈ m l . Recall that |p l (x) − x| < γ −l . We then get Taking l → ∞ and a l < γ −l gives that |A Σ l (p)| 2 → ∞ and proves (A) of Theorem 1.

Proof of Theorem 1 (D) and The Structure Of The Limit Lamination
Lemma 13. A subsequence of the embeddings F k : Ω k → R 3 converges to a minimal lamination of C \ M Proof. Let K be a compact subset of the interior of Ω ∞ . Then for z ∈ K, we have that sup k | d dz H k (z)| < ∞. Montel's theorem then gives a subsequence converging smoothly to a holomorphic function on K. By continuity of integration this gives that the embeddings F k : K → R 3 converge smoothly to a limiting embedding. Thus the surfaces Σ k converge to a limit lamination of C \ M that is smooth away from the M.
Let I = (t 1 , t 2 ) ⊂ R be an interval of the complement of the M in R and consider Ω I = Ω ∞ ∩ {Re z ∈ I}. Then Ω I is topologically a disk, and by Lemma 13, the surfaces Σ k,I ≡ F k (Ω I ) are contained in {t 1 < x 3 < t 2 } ⊂ R 3 and converge to an embedded minimal disk Σ I . Now, Theorem 1(C) (which we have already established), gives that Σ I consists of two multi-valued graphs Σ 1 I , Σ 2 I away from the x 3 − axis. We will show that each graph Σ j