Annuloids and Δ-wings

Proposition 5.6

The map t : M → ( 0,1 ) is proper and bounded above by a constant c < 1.

Proof

For the family Γ(t) in (5), we imposed condition (4) in order assert, using Theorem 1.1, that for t sufficiently close to 1, Γ(t) bounds no connected translator in the halfspace {z ≥ 0}. Therefore, there is a T ∈ (0, 1) such that if M is a symmetric translating annulus with Π(M) = Γ(t), then t < T. Consequently

lies in a compact set K. By Lemma 7.4 (in Subsection 7), it follows that ⋃ M ∈ M M also lies in a compact region of R ³. Using the curvature and area estimates in Remark 5.2 (2), it follows that the map t : M → ( 0,1 ) is proper.

□

Theorem 5.7

Let M i ∈ C be a sequence with ∂ _inner M _i and ∂ _outer M _i , the boundary components of M _i , both smooth curves with nonvanishing curvature. Suppose they converge in C ^2,α to the same smooth convex curve Γ. Denote by σ(M _i ) the length of the shortest homotopically nontrivial curve in M _i .

1. If σ(M _i ) → 0, then M _i converges to D(Γ), and the convergence is C ^2,α away from Γ ∪ (Z ∩ D(Γ)), with multiplicity 2.

2. If inf _i σ(M _i ) > 0, then M _i converges to Γ. For large i, M _i is the graph of a function u _i on the annular region between ∂ _inner M _i and ∂ _outer M _i . Furthermore,

   sup | D u i | → 0 .

Proposition 5.8

There exists a δ > 0 such that if t ∈ (0, δ) then a g-area-minimizing translator with boundary equal to Γ(t) is a graph over the planar annulus between Γ_in(t) and Γ_out(t). Moreover, there is a unique such translating surface M(t) with these properties (graphical or g-area-minimizing) and the M(t) converge as sets to Γ.

Proof

Denote by Ω_out and Ω_in the planar domains bounded by Γ_out and Γ_in, respectively Suppose t _i → 0 and M(t _i ) is a sequence of g-area-minimizing annuli with ∂M _i = Γ(t _i ). Then

Note that the right-hand side of the above inequality converges to 0 as t _i → 0. Hence, area _g (M(t _i )) converges to zero. This implies that the M _i converge to Γ (in Hausdorff distance). If not it would violate the monotonicity formula. This means we may use statement (i) above to conclude that for t _i suitably small, M(t _i ) is a graph. One can now use the maximum principle to show that there is only one graphical translator with the same boundary as M(t _i ) □

Definition 5.9

We define R to be the space of compact, translating annuli with the following properties:

1. ∂ _inner M are ∂ _outer M are rectangles whose sides are parallel to the coordinate axes, and

2. M is the limit of a sequence of translators M n ∈ C such that, for each n, ∂ _inner M _n and ∂ _outer M _n are smooth with nowhere vanishing curvature.

Theorem 5.10

For a, b > 0, let Γ _a,b be the symmetric rectangle that is the boundary of [−a, a] × [−b, b] ⊂ R ². Then for every x ∈ (0, a), there exists a compact annular translator M _a,b (x) whose necksize is x and whose boundary consists of two rectangles Γ _a,b and Γ _A,B , where A ∈ (a, ∞) and B ∈ (b, b + π).

Definition 6.4

Let F be a foliation of an open subset W of a Riemannian 3-manifold N by minimal surfaces. For a minimal surface M ⊂ N, a critical point of M with respect to F is an interior point p where a leaf of F is tangent to M but M is not equal to that leaf in a neighborhood of p. The multiplicity of the critical point is the order of contact of M with the leaf. We denote by

the total number of interior critical points, counting multiplicity.

Definition 6.5

A minimal foliation function is a continuous function F from an open subset W ⊂ N to an open interval I ⊂ R such that

1. F ⁻¹(t) is a minimal surface.

2. F ⁻¹(t) is in the closures of {F > t} and {F < t}.

We define

where F is the foliation whose leaves are the level sets of F .

Theorem 6.6

([15, Theorem 4]). Suppose F : W ⊂ N → I is a proper minimal foliation function on an open domain W of a Riemannian manifold N and that M ⊂ N is a minimal surface with finite genus. Assume that (∂M) ∩{F < t} is empty for some t ∈ I. Let

1. Q equal the set of local minima of F|∂M (presumed to be finite).

2. A equal the set of local minima of F|∂M that are not local minima of F|M.

3. χ(M ∩ W) be the Euler characteristic of M ∩ W.

Then

where |⋅| denotes the number of elements in a set.

Theorem 6.7

Suppose that M _i are minimal surfaces in a 3-manifold N and that M _i converges smoothly to a minimal surface M. Suppose F _i are minimal foliation functions defined on open subsets W _i ⊂ N such that that the F _i converge smoothly to a minimal-foliation function F defined on an open subset W of N. Then

In particular, if p is a critical point of (F, M), then p is a limit of critical points p _i of (F _i , M _i ).

Proposition 6.8

The compact translating annuli M = M _a,b (x) of Theorem 5.10 and the complete embedded annuloids M ∈ A of Theorem 2.3 that are their limits satisfy

1. For each horizontal unit vector v, the function F _v in (9) satisfies

   N ( F v | M ) ≤ 2 .

2. If H is as in (10), then

   N ( H | M ) ≤ 8 .

Proof

We prove the easier case (i) here. For the other case, see Theorem 6.4 in [6]. A compact annulus M in C is bounded by a pair of strictly convex, nested and symmetric curves. By the definition of strict convexity, on each one of the boundary curves there is precisely one local minimum of F _v . Since χ(M) = 0, it follows from (8) that

We may use the lower-semicontinuity property of Theorem 6.7 to assert the same estimate if M is bounded by rectangles as in Theorem 5.10. Now taking a limit of compact annular translators with rectangular boundary curves, and using Theorem 6.7 again, we get the same estimate for annuloids that are limits of these sorts or compact annuli, as in Proposition 6.1 and Theorem 6.3. (Recall that we have the curvature and area estimates needed in Remark 5.2.)□

Theorem 6.9

([6, Theorem 5.4]). Suppose M is a properly embedded translator for which we have estimates (i) and (ii) of Proposition 6.8 and the area and curvature estimates of Remark 5.2. Suppose p _i is a sequence in M that diverges in R ³. Then the sequence of surfaces M − p _i has a subsequence that converges smoothly to a limit surface M′ that is the union of vertical planes and translating graphs. If p _i = (0, 0, z _i ), then the limit surface is the union of vertical planes.

Proof

We may assume that M is connected and not a vertical plane. By hypothesis F _v has no more than two critical points. Let U be the set of critical point of F _v |M and define

Since the p _i diverge, the curvature and area estimates of Remark 5.2 guarantee that the M _i converge smoothly to a limit translator M′. By lower semicontinuity,

Therefore

Let Σ be a component of M′ that is not a vertical plane. From the equality above we may conclude that N ( F v | Σ ) = 0 for any horizontal unit vector v. In other words, the tangent plane to Σ is never vertical. It now follows from Lemma 6.10 below that Σ is a graph. (Alternatively, Corollary 1.2 in [5] implies that Σ is a graph.) Now let W be the set of critical points of H on M. Recall that the level surfaces of H are all graphs of the same type over the same strip in the plane {z = 0}. For a divergent sequence of points p _i = (0, 0, z _i ) a parallel argument to the one in the previous paragraph (with W instead of U) shows that the sequence of surfaces

has a subsequence that converges smoothly to a limit translator M′ with

Suppose that a component Σ of M′ is a graph. This contradicts Lemma 6.11 below, which asserts that for a translating graph G, we can find a grim reaper surface or tilted grim reaper surface so that its associated minimal foliation function H (10) satisfies

□

Lemma 6.10

Suppose Σ is a connected and properly embedded surface lying in a convex domain of W ⊂ R ³ with the property that W\Σ has two connected components. If the tangent planes to Σ are never vertical, then Σ is a graph over a horizontal plane.

Proof

By assumption, the projection

is locally one-to-one. If it is not globally one-to-one, there exists a vertical line L that intersects Σ in two or more points. Choose p and q in L ∩Σ with the property that the interval l ⊂ L between q and p contains no other points in L ∩Σ. Relabel the points if necessary so that q is above p, i.e. q ⋅e ₃ > p ⋅e ₃. Now, by assumption, W \Σ has two components. We may orient Σ so that its unit normal ν points into the component that contains the line segment l. Therefore,

Since Σ is connected, there is a path in Σ between p and q. By the intermediate-value theorem, there must be a point on this path where ν ⋅e ₃ = 0. At this point, the tangent plane to Σ is vertical, a contradiction.□

Lemma 6.11

Suppose G is a complete translating graph. Then there is a grim reaper surface for which the associated minimal foliation function H in (10) satisfies

Proof

According to the classification in Theorem 3.4, we may assume that either G is a bowl soliton or, after a rotation, G is a grim reaper surface, a tilted grim reaper surface or a Δ-wing defined over a strip {|y| < b} for some value of b ≥ π/2. Both the Δ-wing and the bowl soliton contain a point where the height is maximized. In these two cases, let H be the minimal foliation function associated with the standard grim reaper surface: h(x, y) = log(cos y). At these maxima, H|G has a critical point. Therefore N ( H | G ) ≥ 1 .

Next, suppose that G is the grim reaper surface. Rotate G by a nonzero angle around the z-axis, Z, to produce G ̃ , and let H be the minimal foliation function associated with vertical translates of G ̃ . At the origin, G ̃ and G are tangent. Hence N ( H | G ) ≥ 1 .

The last case to consider is when G is a tilted grim reaper surface. Observe that the Gaussian image of G is a semi-circular arc in the hemisphere of S ² ∩{z < 0} with endpoints at (0, ±1, 0) on the equator. This arc does not pass through the point (0, 0 − 1). Now rotate the untilted grim reaper surface by a nonzero angle θ about Z. Its Gaussian image is a semi-circular arc that passes through (0, 0 − 1) and has endpoints at ± (cosθ, sinθ, 0) on the equator, for some θ ≠ 0. These two semi-circular arcs cross. Therefore, there is a translate of the this rotated grim reaper surface that is tangent to G at some point. Hence, if H is the minimal foliation function associated with the translated and rotated grim reaper, we must have, again, N ( H | G ) ≥ 1 .□

Proposition 7.1

([6, Theorem 18.1]). Suppose that I is an infinite open strip in R ² = {z = 0} of width π. Suppose that M is a properly immersed translator in R ³ ∩{z ≥ 0} with no boundary in the slab S ≔ I × R. Then M lies in the complement of S.

Proof

We may assume that the strip I is R × (−π/2, π/2). Fix β ∈ (0, π/2). Define the substrip I _β ≔ R × (−β, β) ⊂ I and, for L > 0, the rectangle R _L,β ≔ [−L, L] × [−β, β]. As discussed in Section 4, there is a unique graphical translator over R _L,β with zero boundary values. By straightforward applications of the maximum principle,

1. M ∩ (R _L,β × R) lies above the graphical translator over R _L,β .

2. If L′ > L, the graphical translator over R _L′,β lies above the graphical translator over R _L,β .

   Vertically translate by −log cos(β) the standard grim reaper surface G, the graph of log(cos(y)) over I. Restrict attention to {z ≥ 0} to produce a graph G _β over I _β . Note that G _β is graph with zero boundary values on I _β . Using the maximum principle again,

3. For any L > 0, the graph G _β lies above the graphical translator over R _L,β .

   From (ii) and (iii) above, we may conclude that the graphs over R _L,β converge smoothly, as L → ∞, to a graphical translator V _β ⊂ {z ≥ 0} over I _β . This graph is zero on ∂I _β . Moreover, from (i),

   (11) M ∩ ( I β × R ) lies above V β .

   This is true for every β ∈ (0, π/2). We claim that V _β = G _β . Assuming the claim we can conclude the proof of the proposition from (11) by observing that a point p = (x, y, z) ∈ S = I × R is below G _β provided |y| < β and β is sufficiently close to π/2.

   To prove the claim we will use the following proposition.

Proposition 7.2

([2, Proposition 3.2]). Let Ω be a bounded, convex domain in R ^p . Suppose there is bounded translating graph W over R ^q × Ω that vanishes on the boundary. Then W is unique, and therefore is invariant under translation in the first q coordinates translation in the depends only on the second p coordinates.

Corollary 7.3

([6, Corollary 18.3]). If M ⊂ {z ≥ 0} is a translator, then M lies in C × [0, ∞), where C is the convex hull of the projection of ∂M to the horizontal plane.

Lemma 7.4

([6, Lemma 18.4]). Let M ⊂ R ³ ∩{z ≥ 0} be a translator such that ∂M lies in a compact set K. Then M is bounded above. In particular, if K lies below a bowl soliton Q, then M also lies below Q.

Theorem

Let U n ⊂ U n ′ be nested, open, convex regions in R ² such that U _n converges to a bounded, open, convex set U and such that U n ′ converges to an infinite strip U′. Suppose that

Then for all sufficiently large n, there is no connected translator in {z ≥ 0} whose boundary is S n ≔ ( ( ∂ U n ) ∪ ∂ U n ′ ) × { 0 } .

Proof

Suppose the result is false. Then (after passing to a subsequence) each S _n bounds a connected translator M _n in {z ≥ 0}. Passing to a further subsequence, the M _n converge as sets to a limit set M. Note that U′\U contains two parallel infinite strips I ₁ and I ₂ each of width π. Thus by Proposition 7.1, M is disjoint from I ₁ × R and I ₂ × R. Hence, M is the union of three connected components, where one component, M*, has boundary (∂U) × {0}, and where each of the other two components is bounded by one of the straight lines in (∂U′) × {0}.

By Lemma 7.4, M* is compact. Let K be a compact set such that M* is in the interior of K and such that M \ M* is disjoint from K. For all sufficiently large n, M _n contains a point in ∂K, and therefore M ∩ ∂K is nonempty, a contradiction.□

Proposition 7.5

Given b ∈ (0, ∞), there is a λ < ∞ with the following property. If M is a translator in {z ≥ 0} with boundary ∂M contained in

then z ≤ λx for all (x, y, z) ∈ M.

Proof of Proposition 7.5

By Corollary 7.3,

Let B > b and consider the rectangular box [0, B] × [−B, B] × [0, B]. Let Q be the polyhedral surface consisting of the x = 0 face and the y = ±B faces of the box. Let w : R ² → R be a bowl soliton. By adding a constant to w, we can assume that ∂Q lies below the graph of u. Let S be a surface (locally integral current, say) in the region

with ∂S = ∂Q that minimizes area with respect to the translator metric. Then S is a translator that is smooth except perhaps at the corners of ∂Q. By Lemma 7.4, S has compact support. By the maximum principle, S is not tangent to ∂K at any non-corner point of ∂S.

Thus there is an ϵ such that S contains the graph of a smooth function

with ∂ u ∂ x > 0 at all points in its domain. Hence there is a λ > 0 for which

We claim that z ≤ λx for all x ∈ M. Suppose not. Then there exists x ̂ ∈ [ 0 , ∞ ) with

Let

Then M ̂ satisfies the hypotheses of the Proposition, and

Furthermore, by Corollary 7.3,

If S + (0, 0, ζ) intersected M ̂ for some ζ ≥ 0, then there would be a smallest x ≤ 0 such that S′ ≔ S + (x, 0, ζ) intersected M ̂ , and the strong maximum principle would be violated at the point of contact. (It follows from (13) that the point of contact would be at an interior point of M ̂ and of S′.) Thus S + (0, 0, z) is disjoint from M for all z ≥ 0.

It follow that M ̂ ∩ { x ≤ ϵ } lies below the graph of u and thus that

But this contradicts (12).□

Corollary 7.6

If M is a translator in {z ≥ c} with boundary contained in

then z ≤ c + λ(x − a) for all (x, y, z) ∈ M.

Proposition 7.7

Given B, there is a C = C _B with the following property. If M is a translator in M ∩{x ≥ 0}∩{|y| ≤ B} with ∂M ⊂ {x = 0}, then for 0 ≤ x ≤ x′,

Proof of Proposition 7.7

Note that the desired inequality is valid if and only if it is valid for all vertical translates of M. First, let x = 0. If z∗(M, 0) = ∞, there is nothing to prove. Otherwise, we may assume, without loss of generality, that z∗(M, 0) = 0. Let M′ = M ∩{z ≥ 0}. Note that M′ satisfies the assumptions of Proposition 7.5, so the desired inequality is satisfied with C = λ. We now know that M ∩{x = x′} is bounded above for any value of x′. Repeating the argument above for 0 < x′ ≤ x″ completes the proof.□

Proposition 7.8

Fix β ∈ (0, π/2) and let R _L = R _L,β = (−L, L) × (−β, β). There exists a length L(β) and a translator

such that

The graph of v _β is bounded by the vertical lines passing through the four corners of R _L(β),β . By adding a constant to v _β , we may assume that v _β (0, 0) = 0. Furthermore, as β → π/2, L(β) → ∞ and v _β → u, the grim reaper surface

Theorem

Let M be a connected, properly embedded translator in {z ≥ 0} bounded by two parallel lines y = ±b in the plane z = 0. Suppose that either M ∩{x = 0} is bounded or that M is simply connected. Then M is a portion of a grim reaper surface. That is, b < π/2 and M is the graph of the function

Proof

First we observe (by Corollary 7.3) that

and that b < π/2. We will prove the theorem by showing that M lies in the closed region above the graph of u − log(cos b) and also in the closed region below the graph of u − log(cos b).

Claim 1

If M ∩{x = 0} is bounded, then there is a λ < ∞ such that

Proof of Claim 1

Let h = sup{z : (0, y, z) ∈ M}. By hypothesis, h < ∞. By Proposiiton 7.5 applied to (M − h) ∩{z ≥ 0}∩{x ≥ 0} and to (M − h) ∩{z ≥ 0}∩{x ≤ 0}, we see that we have the the linear bound (15).□

Claim 2

If M is simply connected then we have the linear bound (15).

Proof of Claim 2

Because M is simply connected, a curve γ in M that begins on one of the boundary lines and ends on the other divides M into two components. Each component is bounded by γ together with two divergent rays, one in each of the boundary lines. Let M ₊ be the component whose boundary contains rays diverging in the positve x-direction, and let M ₋ be the other one. Denote by C ₊ be the convex hull of the projection of ∂M ₊ onto the strip {|y| ≤ b}∩{z = 0}. By Lemma 7.4, M ₊ lies in C ₊ × [0, ∞]. For suitably small a ∈ R, C ₊ ⊂ x ≥ a × {|y| ≤ b}∩{z = 0}. Hence we may apply Proposition 7.5 to conclude, as in Claim 1, that M ₊ satisfies (15). An analogous argument works for M ₋.□

Theorem

Suppose M is a properly embedded and connected translator that lies in a vertical slab {|y| < B}. If there exists a constant a such that M ∩{x = a} is bounded above, then B ≥ π.

Proof

Let M* = M ∩{x ≥ a}. By hypothesis, there exists a real number, c, such that M* ≔ M ∩{x ≥ a}∩{z ≥ c} satisfies the hypothesis of Corollary 7.6, namely that ∂M* ⊂ [a, ∞) × [−B, B] × {c}. Therefore,

for all (x, y, z) ∈ M*.

Suppose B < π/2. Translate M* horizontally, if necessary, so that a = 0. For any β satisfying B < β < π/2, we may find a Scherk translator v _β defined on the rectangle R _L,β = (−L, L) × (−β, β), (where L = L(β)) with the property that v _β equals −∞ on the horizontal sides of the boundary of R _L,β and +∞ on vertical sides. Since the height of M* ∩{0 ≤ x ≤ L} is bounded above by c + λx, there exists a constant d so that v _β + d lies above M* ∩{0 ≤ x ≤ L}. This violates the maximum principle because the boundary of the Scherk translator (see Proposition 7.8) consists of the four vertical lines through the corners of R _L,β , and β > B.□