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Properly embedded surfaces with constant mean curvature

Antonio RosHarold Rosenberg

American Journal of Mathematics, Volume 132, Number 6, December2010, pp. 1429-1443 (Article)

Published by The Johns Hopkins University Press

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PROPERLY EMBEDDED SURFACESWITH CONSTANT MEAN CURVATURE

By ANTONIO ROS and HAROLD ROSENBERG

Abstract. In this paper we prove a maximum principle at infinity for properly embedded surfaceswith constant mean curvature H > 0 in the 3-dimensional Euclidean space. We show that no oneof these surfaces can lie in the mean convex side of another properly embedded H surface. We alsoprove that, under natural assumptions, if the surface lies in the slab |x3| < 1/2H and is symmetricwith respect to the plane x3 = 0, then it intersects this plane in a countable union of strictly convexclosed curves.

1. Introduction. In this paper we derive some global properties of properlyembedded surfaces in R3 of nonzero constant mean curvature H. We call sucha surface an H-surface. Our main result is a maximum principle at infinity forthese H-surfaces.

THEOREM 1. Let M1 and M2 be connected disjoint H-surfaces in R3. Then M2

is not on the mean convex side of M1.

The surface M1 separates R3 into two connected components since M1 isproperly embedded. The mean convex side of M1 is the component W1 of R3−M1,towards which points the mean curvature vector of M1.

In the minimal case, the Halfspace Theorem implies that two disjoint properimmersed minimal surfaces must be parallel planes, see [8].

Assume H = 0 and M2 ⊂ W1. If there exist points x ∈ M1 and y ∈ M2

whose distance is dist(M1, M2), then the theorem above can be proved directly asfollows. There are no focal points of M1 along the interior of the line segmentfrom x to y, since the segment minimizes distance between M1 and M2, and aftera small translation of M2 we can assume that there are not focal points of M1

in the segment xy. So the equidistant surfaces to M1 are non-singular along thissegment, starting with a small neighborhood of x on M1. Their mean curvatureis strictly increasing when one goes from x to y. But the surface at y touches M2

at its mean convex side, a contradiction.If the above points x and y do not exist, we can take divergent sequences

xn ∈ M1, yn ∈ M2 such that |xn − yn| → dist(M1, M2) and xn − yn converge to a

Manuscript received April 25, 2006; revised November 26, 2009.Research of the first author supported in part by MEC-FEDER MTM2007-61775 and J. Andalucıa

P09-FQM-5088 grants.American Journal of Mathematics 132 (2010), 1429–1443. c© 2010 by The Johns Hopkins University Press.

1429

1430 ANTONIO ROS AND HAROLD ROSENBERG

vector v. So the surface M2 +v lies in the mean convex side of M1, M2 +v ⊂ W1,and touches M1 at infinity, that is, dist(M1, M2 + v) = 0. Theorem 1 says that thiscan not happen, which explains why we call it the maximum principle at infinity.

We will also study H-surfaces M in a slab of R3; between two horizontalplanes say. An unsolved problem is whether such a surface admits a horizon-tal plane of symmetry. It is natural to attack this problem by the technique ofAlexandrov reflection [1], starting with horizontal planes coming down from thetop of the slab. This method requires that the part of surface above the movingplane be a graph over a horizontal planar domain and that its reflected image withrespect to this plane lies in the mean convex side W of M. If there is a first planewhere the symmetry of M above the plane touches the part of M below, then theusual maximum principle says M is symmetric about this plane. In the case M iscompact, we deduce in this way that the surface has a lot of mirror symmetriesand so it must be a round sphere, [1]. However in our setting this does not appearto work for several reasons. First, one might not be able to get started. In fact,if we suppose M has unbounded curvature near the plane x3 = a, a being thesupremum of x3 restricted to M, then moving this plane down till x3 = a− ε, forsmall ε > 0, the part of M above this new plane may not be a vertical graph. Ifone assumes M has bounded curvature then this phenomena does not occur, so atleast one can begin to do symmetry of the part of M above the plane. Even then,one quickly encounters another difficulty: it may be the case (and Pascal Collinhas constructed H-surfaces with boundary doing this [2]) that at the last allowedposition of the moving plane, the part of M below this plane and the symmetryof the part above touch for the first time at infinity. Then we can not proceed.

This leads us to study the class S of (non-necessarily connected) properlyembedded H-surfaces in R3 satisfying the following conditions: M lies in a hor-izontal slab, is symmetric about the plane P = x3 = 0 and M+ = M ∩ x3 > 0is a graph over a open set in P. This is the class of surfaces we would obtainif Alexandrov reflection technique could be applied to proper H-surfaces in aslab. There are many such examples. The Delaunay surfaces are in S and theirwidth varies from 2/H (attained for the limit case of a stack of spheres) to 1/H(for the cylinder). Kapouleas [9] has constructed examples of finite topology inS which look like a sphere with n horizontal Delaunay ends, the directions ofthe ends symmetrically placed, like the n roots of unity. For further results onsurfaces in S with finite topology see Grosse-Brauckmann, Kusner and Sullivan[7]. Lawson [11] constructed doubly periodic H-surfaces in S and Ritore [17]and Grosse-Brauckmann [6] have given more doubly periodic examples of thistype. In particular, Ritore constructs examples whose width tends to zero (for Hfixed): these surfaces look like two close parallel planes connected by a doublyperiodic family of catenoidal necks, the distance between two neighbor necksbeing small. Notice that each surface in S has width at most 2/H by Theorem 6.

We will give a structure result for H-surfaces in the class S contained betweentwo parallel planes at distance smaller than 1/H: Assuming that M has bounded

PROPERLY EMBEDDED SURFACES WITH CONSTANT MEAN CURVATURE 1431

curvature, we prove that the shape of the surface is close to the doubly periodicH-surfaces constructed by Lawson [11].

2. A priori estimates for H-surfaces. In this section we give a prioriestimates for H-surfaces in different contexts. These results will be used in theproof of our main results in the sections §3 and §4 below. We give a bound,by modifying arguments of Fisher-Colbrie [3], for the radius of a geodesic ballcontained in the interior of a stable H-surface and we extend Serrin’s heightestimate for compact H-graphs, [18], to H-graphs over arbitrary domains.

Let M be an H-surface in R3.We will consider on M the unit normal vectorfield n which makes H > 0. Equivalently, n will be the normalized mean curvaturevector.

An H-surface M is said to be stable (in the strong sense) if for any functionu with compact support in M we have that

∫M |∇u|2−|A|2u2 ≥ 0, where ∇u and

A denote the gradient of u and the shape operator of M respectively. Stability isequivalent to the existence of a positive solution of the equation ∆v + |A|2v = 0 onM, where ∆ denotes the Laplacian of M, see [4]. In particular any graph is stable,because the third coordinate of the unit normal vector n3 satisfies the equationabove (for H-surfaces there is another natural notion of stability, weaker than theone considered in this paper, which is related with the isoperimetric problem: weask that the integral inequality holds for any u with

∫M u = 0).

A fundamental fact about stable H-surfaces is that we have an estimate (de-pending only on H) of the largest geodesic ball contained in the interior of thesurface. The proof of the next result follows from the ideas of Fischer-Colbrie[3] and it is implicit in Lopez and Ros [12].

THEOREM 2. Let M be a stable H-surface. Then the (intrinsic) distance of anypoint of M to ∂M is smaller than or equal to π/H.

Proof. The operator L = ∆ + |A|2 = ∆ + (4H2 − 2K), K being the Gausscurvature of M, has index zero and, so, there is a positive function u on M withLu = 0, (see Proposition 1 of Fisher-Colbrie [3]).

Consider the new metric ds2 = u2 ds2 and let p ∈ M and R > 0 be such thatthe open ds2-geodesic ball B = B(R, p) ⊂ M centered at p is relatively compactin M. It is enough to prove that R ≤ π/H.

Let γ be a minimizing geodesic for the metric ds2, joining p to ∂B. As theds2-distance between p and any point of ∂B is R, then if we denote by a theds2-length of γ we have a ≥ R. Parameterize γ by arclength s in the ds2 metric,0 ≤ s ≤ a.

Since γ is minimizing for ds2, the second variation of length yields:

0 ≤∫ R

0

((dφds

)2

− Kφ2

)ds,(1)


for all φ with φ(0) = φ(R) = 0, R being the ds2-length of γ and K the Gausscurvature of the metric ds2.

We have that

dφds

=dφds

dsds

=1u

dφds

and

K =1u2 (K − ∆ log u).

Moreover, as u is a Jacobi function on B, we can write

0 = Lu = ∆u− Ku +(

2H2 +|A|2

2

)u ≥ ∆u− Ku + 2H2u.

Therefore, if c = 2H2 and u′(s) = d(u γ)/ds, we obtain

∆ log u =u∆u− |∇u|2

u2 ≤ K − c− (u′)2

u2(2)

where we have used that |u′| ≤ |∇u|.Then (1) and (2) yield:

∫ a

0

(cu

+(u′)2

u3

)φ2 ds ≤

∫ a

0

K − ∆ log uu

φ2 ds ≤∫ a

0

1u

(φ′)2 ds.(3)

Write φ = uψ, where ψ(0) = ψ(a) = 0, so that

φ′ = u′ψ + uψ′ and1u

(φ′)2 =1u

(u′)2 ψ2 + u(ψ′)2 + 2u′ψψ′.

Then (3) yields:

∫ a

0

(c uψ2 +

(u′)2

uψ2

)ds ≤

∫ a

0

((u′)2

uψ2 + uψ′ + 2u′ψψ′

)ds

An integration by parts using d(uψψ′) = (u′ψψ′+u(ψ′)2 +uψψ′′)ds transformsthis last inequality to: ∫ a

0(cψ2 + (ψ′)2 + 2ψ′′ψ)u ds ≤ 0.(4)

Finally, we take

ψ(s) = sin(πsa

), 0 ≤ s ≤ a,


and so (4) becomes:

∫ a

0

(π2

a2 cos2(πsa

)+(

c− 2π2

a2

)sin2

(πsa

))u(γ(s)) ds ≤ 0.

Thus c < 2π2

a2 and therefore R ≤ a < πH , which proves the theorem.

Now we consider H-surfaces M which are vertical graphs (in short, H-graphs). The unit normal vector of a graph is never horizontal and so, on eachconnected component of M, it points either down or up. Recall also that H-graphsare stable.

LEMMA 3. Let M be an H-surface given as the graph of a smooth positivefunction u defined over a domain Ω ⊂ x3 = 0. If M is properly embedded in thehalfspace x3 > 0 then,

(a) M is contained in the slab 0 < x3 < 2π/H,(b) u extends continuously to zero on the boundary of Ω, and(c) the normal vector of M points down.

Proof. As any H-graph is stable, it follows from Theorem 2 that x3 < 2π/Hon M: Otherwise we can find a geodesic ball of radius π/H contained in the inte-rior of M. This proves (a). In particular, we conclude that u extends continuously,with zero boundary values, to the topological boundary of Ω. So it remains toprove (c). Take a vertical line l which intersects a connected component M′ ofM and a sphere S of mean curvature H centered at a high point of l so that thesphere is disjoint from M and l ∩ S is above l ∩M. Move S down till we havea first contact with M′. If the mean curvature vector of M′ points up, then themaximum principle would imply that the graph M contains a sphere, which isimpossible.

From the Lemma above we conclude that for an H-graph M ⊂ x3 > 0, Mis properly embedded in the upper halfspace if and only if u extends continuouslyto zero on the boundary of Ω.

We will consider limits of H-graphs. The following propositions justify theexistence of such limits. First we prove that an H-graph M in the upper halfspacewith zero boundary values satisfies an interior curvature estimate in x3 > 0.This is a known fact but we include a proof for the sake of completeness.

PROPOSITION 4. There is a positive constant C, depending only on H > 0, suchthat any H-surface M properly embedded in x3 > 0 given as the graph of afunction u ∈ C∞(Ω), with Ω ⊂ x3 = 0, satisfies

|A| ≤ Cx3

,

|A| being the length of the second fundamental form of M.


Proof. This can be shown as follows: In case the estimate fails, we canconsider a sequence pk, k = 1, 2, . . ., of points in the surfaces Mk, satisfying thehypothesis of the assertion such that |Ak(pk)|x3(pk) > k, where Ak is the secondfundamental form of of Mk. Let B(p, r) = x ∈ R3/|x−p| < r and rk = x3(pk)/2.It follows that for q ∈ Mk ∩B(pk, rk), the expression (rk− |q− pk|)|Ak(q)| attainsits maximum at a point qk (as it vanishes when q approaches to the boundary).Moreover if r′k = rk − |qk − pk|, then we have Sk = Mk ∩ B(qk, r′k) ⊂ Mk ∩B(pk, rk) and Rk = r′k|Ak(qk)| > k/2. These points qk are called points of almostmaximal curvature and the translated rescaled surfaces Σk = |Ak(qk)|(Sk − qk)converge, up to a subsequence, to a nonflat complete surface Σ∞ in the Euclidean3-space with mean curvature 0. To see that we observe that the surfaces Σk passthrough the origin and ∂Σk is contained in the boundary of the ball B(0, Rk) whoseradius Rk converges to ∞. Moreover in the ball B(0, Rk/2) the length the secondfundamental form of Σk is bounded by 4, and equals 1 at the origin. The meancurvature of Σk is equal to H/|Ak(qk)|.

This permits one to construct a subsequence of Σk that converges to a com-plete minimal surfaces Σ∞ passing through the origin and with curvature 1 at theorigin.

As the Gauss map of Σ∞ lies in the closed lower hemisphere, it followsthat the minimal surface Σ∞ is stable and therefore it must be flat, see [5]. Thiscontradiction proves the assertion.

PROPOSITION 5. Let Mn be a sequence of H-surfaces, H > 0, such that Mn

is the graph of a positive function un over an open subset Ωn ⊂ x3 = 0 thatextends continuously to zero on the boundary of Ωn. Assume that there are pointspn ∈ Mn with pn −→ p and x3(p) > 0. Then there exists a subsequence of Mnwhich converges on compact subsets of x3 > 0 to the graph of a positive functionu : Ω −→ R, over an open subset Ω, which extends continuously to the closure ofΩ with zero boundary values.

Proof. The curvature estimate in the Proposition above implies that, up to asubsequence, the surfaces Mn converge to an H-surface M properly immersed inx3 > 0. If n3 denotes the third coordinate of the unit normal vector of M, wehave that ∆n3 + |A|2n3 = 0. As M is a limit of graphs we get from item c) inLemma 3 that n3 ≤ 0 and the maximum principle implies that either n3 < 0 inM or n3 = 0 on a connected component of M.

In the second case, that component must be a vertical circular cylinder in-tersected with the upper halfspace. By lemma 3, the height of the surfaces Mn isuniformly bounded, so this case is impossible.

Therefore n3 < 0 and we claim that M is the graph of a positive functionu over an open subset Ω ⊂ x3 = 0. To prove this claim note that each pointp ∈ M has a neighborhood Up which is the graph of a smooth function over anopen disc in the plane x3 = 0. If there were two different points p, q ∈ M lying inthe same vertical line, we deduce that the same would be true for the graph Mn,


for n large enough. This contradiction proves that M meets each vertical line atmost once and so M is a graph of a function u. From Lemma 3 it follows that uextends continuously to the closure of Ω with zero boundary values. That provesthe proposition.

If is well-known that compact H-graphs with zero boundary values satisfya height estimate, see Serrin [18]. In the result below we extend this fact toarbitrary H-graphs with that boundary condition.

THEOREM 6. Let M be an H-surface, H > 0, given by the graph of a positivefunction u on a planar domain Ω ⊂ x3 = 0, where u extends continuouslyto the closure of Ω with zero boundary values. Then M is contained in the slab0 < x3 ≤ 1/H.

Proof. First observe that Theorem 2 implies that x3 is bounded on M. Toprove the upper bound x3 ≤ 1/H consider on M the function

φM = φ = H x3 + n3,

where x3 and n3 denote the third coordinate of the position and unit normalvectors on M (note that n3 < 0 on M). Then ∆φ = −2(H2 − K)n3 ≥ 0.

If we assume that Ω is smooth and compact, then the statement is a classicalresult of Serrin, [18]: to prove it, observe that φ is subharmonic and it attains itsmaximum at the boundary. On ∂Ω, φ = n3 ≤ 0 and therefore φ ≤ 0 in Ω. ThusH x3 + n3 ≤ 0 which implies

x3 ≤−n3

H≤ 1

H.

Now suppose Ω is noncompact. If φ ≤ 0 on M then we are done, so we cansuppose that supφ = c > 0. Let pn ∈ M be a sequence with φ(pn)→ c, x3(pn)→x∞, and n3(pn) → n∞. Notice that x∞ > 0, since if x∞ ≤ 0, c = Hx∞ + n∞ ≤ 0.

Let Mn be the horizontal translate of M which places pn on the x3-axis,intersected with the open half space x3 > 0. From Proposition 5 we havethat a subsequence (that we also denote by Mn) converges to an H-graph whichis properly embedded in x3 > 0. Let p∞ = lim pn and M∞ the connectedcomponent of the limit of Mn which contains the point p∞. As x3(p∞) = x∞ > 0,we see that p∞ is an interior point of M∞. As the function φ∞ = φM∞ achievesits maximum at p∞ we conclude from the maximum principle that φ∞ is constanton M∞, which means that M∞ is a spherical cap. In particular φ∞ ≤ 0 whichcontradicts that φ∞(p∞) = c > 0.

The lower bound 0 < x3 is proved in a similar way using the equation∆x3 = 2Hn3 < 0. If x3 is negative somewhere on M, then a suitable sequence of


horizontal translated images of M will converge in x3 < 0 to an H-graph M′∞properly embedded in the lower halfspace x3 < 0, whose unit normal vectorpoints down and such that x3 attains its minimum at the interior, which contradictsthe maximum principle. Thus x3 ≥ 0 on M and the maximum principle againgives that x3 > 0, as we claimed.

3. The maximum principle at infinity. In this section we will prove ourmain result: Let M1 and M2 be two connected properly embedded H-surfaces inR3 (H > 0) such that M2 lies in the mean convex side of M1. Then we want toshow that M2 = M1.

By the comments in the introduction, we can assume that M1 is noncompact.We orient M1 and M2 by unit vector fields N1 and N2 whose direction is thatof the mean curvature. The mean convex side W1 of M1 is the component ofR

3−M1 such that N1 along M1 points into W1. So M2 is contained in the closureof W1.

Assuming that M2 = M1 we will obtain a contradiction. Clearly M2 ∩M1 = ∅by the usual maximum principle. Denote by W the component of W1 − M2

satisfying ∂W = M1 ∪ M2. Thus the boundary of W is not connected and themean curvature vector of M1 points to W (hence N1 as well). If W were meanconvex, then the halfspace theorem [8] would imply that M1 and M2 are parallelplanes. Therefore the mean curvature vector of M2 (hence N2 as well) pointsaway from W.

Let S be a relatively compact domain in M1 with smooth boundary Γ. Assumealso that S is unstable. We will show that there is a surface Σ ⊂ W bounded byΓ with constant mean curvature H that is stable. Then by taking the domain S inM1 to be larger and larger, we will obtain a contradiction, using the fact that thedistance between a point of a stable H-surface and its boundary is bounded.

Let F be the family of regions Q ⊂ W enclosed by S and surfaces Σ ⊂ Wwith ∂Σ = Γ and define the functional

F(Q) = A(Σ) + 2H V(Q),

where A(Σ) is the area of Σ and V(Q) is the volume of Q. Our goal is to showthat there is Q ∈ F minimizing the functional F and such that the interior of Σis contained in the interior of W. This will imply that Σ is a smooth surface with∂Σ = Γ and constant mean curvature H. Moreover the mean curvature vector ofΣ points outside Q and Σ is stable in the sense of §2.

We will need certain auxiliary surfaces and regions and some preliminaryremarks. Consider the least area surface Smin in W1 spanning the curve Γ andhomologous to S. Smin is a compact smooth minimal surface and by the maximumprinciple we have M1 ∩ Smin = Γ. Denote by Qmin the region in W1 enclosed byS ∪ Smin.


Let S0 = M2 ∩ B(ρ) be a relatively compact open set in M2, where ρ is alarge positive radius and B(ρ) = x ∈ R3/|x| < ρ. There is an ε > 0 such thatthe sets St = x − tN2|x ∈ S0, 0 ≤ t ≤ ε, are smooth surfaces parallel to M2

foliating a neighborhood Qpar of S0 in W. We put Spar = Sε. Denote by Y theunit vector field, normal to the foliation St, and oriented by N2 and let Ht be themean curvature of St. A calculation shows Ht < H for 0 < t. At any point of St

we have div Y = −2Ht, where div is the divergence operator in R3. Therefore

div Y > −2H, for t > 0.(5)

Finally, we consider ϕ the first eigenfunction of the Jacobi operator of S. Soϕ vanishes at Γ, is positive at the interior of S and satisfies ∆ϕ+ |A|2ϕ+λ1ϕ = 0,where λ1 is a negative constant (as S is unstable). After a perturbation of S we canassume that 0 is not an eigenvalue of ∆+ |A|2. Hence there is a smooth function von S, vanishing at Γ and such that ∆v + |A|2v = 1 in S. The boundary maximumprinciple implies that the derivative of ϕ with respect to the outwards pointingnormal vector is negative along Γ. Therefore we conclude that, for a > 0 smallenough, u = ϕ + av is positive at the interior of S.

For small ε > 0 and 0 < t ≤ ε, u defines a normal deformation of S,

S′t = x + tuN1 | x ∈ S ⊂ W.

The surfaces S′t, 0 < t < ε, foliate an open set Quns ⊂ W. Putting Suns = S′ε, wehave ∂Quns = S ∪ Suns, see Figure 3.

If X is the unit normal vector field of the foliation S′t oriented by N1 and H′tis the mean curvature of S′t, we have div X = −2H′t as above. Moreover

ddt |t=0

2H′t = ∆u + |A|2u = −λ1ϕ + a > 0,

which implies, choosing ε small enough, that H′t > H. Therefore

div X < −2H, for 0 < t < ε.(6)

LEMMA 7. Let Q ∈ F be enclosed by S and Σ. Assume that Σ is smooth at theinterior of W.

(i) If Q ⊂ Qmin, then F(Q ∩ Qmin) < F(Q).(ii) If Q ∩ Qpar = ∅, then F(Q− Qpar) < F(Q).(iii) If Quns ⊂ Q, then F(Q ∪ Quns) < F(Q).

Proof. The assertion in (i) follows because Q′ = Q ∩ Qmin ∈ F and satisfiesV(Q′) < V(Q) and A(Σ′) ≤ A(Σ), where Σ′ = ∂Q′ − S, see Figure 1.


Figure 1. If Σ meets the exterior of the least area surface Smin, then F(Q ∩ Qmin) < F(Q).

The claim in (ii) is a consequence of the inequality div Y > −2H on Qpar,see Figure 2. From the divergence theorem we obtain

−2HV(Q ∩ Qpar) <∫

Q∩Qpar

div Y

=∫∂(Q∩Qpar)

〈Y , ν〉 =∫

Q∩Spar

〈Y , ν〉 +∫

Σ∩Qpar

〈Y , ν〉,

where ν is the outer pointing unit normal to the boundary of Q ∩ Qpar. Usingthat ν = −Y along Q∩ Spar and 〈Y , ν〉 ≤ 1 at the points of Σ∩Qpar, we deduce,after rearrangement, that

−2HV(Q ∩ Qpar) + A(Q ∩ Spar) < A(Σ ∩ Qpar),

and, therefore

F(Q− Qpar) = 2H(V(Q)− V(Q ∩ Qpar)

)+ A(Σ− Qpar) + A(Q ∩ Spar)

< 2HV(Q) + A(Σ ∩ Qpar) + A(Σ− Qpar) = F(Q),

as we claimed.To prove (iii) we argue as in the proof of (ii), but here we use that div X <

−2H on Quns, see Figure 3.

−2HV(Quns − Q) >∫

Quns−Qdiv X =

∫∂(Quns−Q)

〈X, ν〉

=∫

Suns−Q〈X, ν〉 +

∫Σ∩Quns

〈X, ν〉.

Figure 2. If Q cuts the tubular neighborhood Qpar of M2 (in gray), then F(Q− Qpar) < F(Q).


Figure 3. If Q does not contain the shadowy region Quns, then F(Q ∪ Quns) < F(Q).

As ν = X on Suns − Q and 〈X, ν〉 ≥ −1 at the other points of the boundary, wehave

2HV(Quns − Q) + A(Suns − Q) < A(Σ ∩ Quns).

Hence

F(Q ∪ Quns) = 2H (V(Q) + V(Quns − Q)) + A(Suns − Q) + A(Σ− Quns)

< 2HV(Q) + A(Σ ∩ Quns) + A(Σ− Quns) = F(Q),

as desired.

Proof of Theorem 1. Assume 0 lies in M1 and let S(r) be the connectedcomponent of M1 ∩ B(r), where r > 0, and B(r) is the euclidean ball of radius rcentered at 0. Since M1 is properly embedded, the boundary of S(r) is containedin ∂B(r). Let γ be a path in W joining 0 to a point y in M2. Choose r largeenough so that γ is contained in B(r) and dist(γ, ∂B(r)) > 2π/H.

From Theorem 2, and our choice of r, we know that S = S(r) is unstable.Let K be the compact connected domain contained in Qmin, bounded by S

and M2 ∩ Qmin, see Figure 4.Let a be the infimum of F on F . If (Qn, Σn) is a minimizing sequence for F

(i.e. limn→∞ F(Qn, Σn) = a) then by Lemma 7, we can cut and paste the Qn toform a new minimizing sequence (Qn, Σn) so that Qn ⊂ K and int Σn ⊂ intK.

By compactness [15, 5.5], there is a minimum Q of F in K bounded by arectifiable current Σ with the support of Σ contained in W, ∂Σ = ∂S, and Σ disjoint

Figure 4.


from M2. Regularity [14, Corollary 3.7] implies the part of Σ in the interior ofK is a smooth surface of mean curvature H. This stable surface Σ intersects γ(since the union of Σ and S bounds Q). However, for z in the intersection of γand Σ, we have dist(z, ∂Σ) > 2π/H. This contradicts Theorem 2 and completesthe proof of Theorem 1.

4. H-surfaces of bounded width. We now consider properly embedded H-surfaces M with ∂M = ∅ and H > 0, which fit between two parallel planes P0 andP1. The width of M is the infimum of the distance between such planes. The onlycompact connected M is the sphere of width 2/H, so the surfaces we consider arenon compact. We know very little about these surfaces, even assuming boundedcurvature. Assuming M is connected, some questions we cannot answer are:

If M has bounded width, is it at most 2/H?

If M is between the planes P0 and P1, is there a parallel plane P between P0

and P1 with M symmetric by P?

We will study a special class of surfaces of bounded width. Let S be those(non-necessarily connected) properly embedded H-surfaces M, H > 0, which aresymmetric with respect to the plane P = x3 = 0 and such that M+ = M∩x3 > 0is a graph over an open subset Ω in P. Theorem 6 implies that each surface inS has width at most 2/H. Among the surfaces in S, the sphere, the Delaunaysurfaces, and Kapouleas [9] finite topology examples have width at least 1/H.Lawson doubly periodic H-surfaces [11] and the related ones constructed byRitore [17] and Grosse-Brauckmann [6] may have arbitrarely small width. Forsome of these doubly periodic surfaces the domain Ω consists of the plane Pwhere we have removed infinitely many pairwise disjoint compact convex disks.In this section we will prove that any surface in S with width smaller than 1/H(= the width of a cylinder of mean curvature H) looks like these doubly periodicsurfaces.

THEOREM 8. Suppose M in S has width less than 1/H and M+ is a graph overthe open subset Ω ⊂ P. Then the components of P − Ω are strictly convex. Inparticular M is connected. If moreover M has bounded curvature, then P−Ω is acountable disjoint union of strictly convex compact disks.

Proof. First we observe that if M were contained in two non-parallel slabs,then the slabs intersect and M is cylindrically bounded. A theorem of Kusner,Korevaar and Solomon [10] would imply that the components of M are spheresor Delaunay surfaces, which contradicts our width assumption. Therefore, M iscontained in the slab perpendicular to P.

The proof uses an operator L that has its origins in a paper by Payne andPhilippin [16] defined on any surface M ∈ S. The operator is of the form

Lf = ∆f + 〈∇f , X〉,


where X is a tangent vector field to M, singular where M is horizontal, and ∇fdenote the gradient of a function f on M.

As we have oriented M so that H > 0, the maximum principle implies thatn3 < 0 in M+. Consider

ψ = 2Hx3 + n3 on M+,(7)

Clearly ψ = 0 on Γ = ∂M+ = P ∩ M since M is vertical along Γ. A simplecalculation shows ∆ψ = 2Kn3, where K is the Gauss curvature of M. Now if welook for a vector field X such that Lψ = 0, then it suffices to find X satisfying〈X,∇ψ〉 = −2Kn3.

Denote by a the tangent part of e3 = (0, 0, 1). Then ∇x3 = a, ∇n3 = −Aa,where A is the shape operator of M, and

∇ψ = 2H∇x3 +∇n3 = (2HI − A)a,

I being the identity tensor. Using the basic matrix equality A2 − 2HA + KI = 0,one has

A∇ψ = (2HA− A2)a = Ka.

Consequently, if one defines

X = −2n3

|a|2 Aa, where a = 0,

we conclude

〈X,∇ψ〉 = −2n3

|a|2 〈Aa,∇ψ〉 = −2n3

|a|2 〈a, A∇ψ〉 = −2n3

|a|2 〈a, Ka〉 = −2Kn3

and ψ is a solution of Lψ = 0 where a = 0. If ν denotes the outer conormal toM+ along Γ, then by direct computation one has

∂ψ

∂ν= kΓ,

where kΓ is the curvature of Γ in P with respect to the plane unit normal pointingtowards P−Ω. In particular if ∂ψ

∂ν > 0, then Γ is strictly convex towards P−Ω.The maximum principle applied to Lψ = 0 yields directly the following

properties:(i) If ψ assumes an extremum at q ∈ M+, then q is on Γ or the unit normal

vector of M at q is vertical (i.e., q is a singular point of X).(ii) If ψ has a local maximum at q ∈ Γ, then either kΓ(q) > 0 or the component

of M passing through q is a cylinder.


As M is vertical along Γ we have that ψ = 0 on Γ. We claim that ψ ≤ 0 onM+. Suppose first that ψ attains its maximum at q. If q /∈ Γ, then by (i) we getthat the normal vector at q is vertical. So n3(q) = −1, and ψ(q) = 2Hx3(q) − 1.Since the width of M is at most 1/H, we have x3(q) < 1/(2H), and ψ(q) < 0,which is impossible. So, q ∈ Γ and then ψ is nonpositive.

If ψ does not attain its maximum, let qn ∈ M+, ψ(qn)→ supψ. If x3(qn)→ 0,then ψ(qn) = 2H x3(qn)+n3(qn) ≤ 2H x3(qn)→ 0, so supψ ≤ 0 and ψ ≤ 0 on M+.So we can assume x(qn) converges to a number c > 0. Translate M horizontallyso that qn is over the origin, and let M∞ be a limit H-surface of the translatedsurfaces in the open halfspace x3 > 0. This surface is non empty because thepoint q∞ obtained as limit of the translated images of qn lies on M∞. Moreoverthe function ψ∞ constructed as in (7) on M∞ attains its maximum at q∞ andψ∞(q∞) > 0. As M∞ is contained in the slab 0 < x3 < 1/(2H), reasoning as inthe first case we obtain a contradiction. Hence ψ ≤ 0 on M+.

Now we can apply property (ii) to conclude that either M has a cylindricalcomponent or kΓ(q) > 0 for all q ∈ Γ. Since we are assuming the width is strictlyless than 1/H, we have kΓ > 0 at each point of Γ. So the connected componentsof P−Ω are strictly convex.

Next we prove that, if M has bounded curvature, then the planar curvature kΓis bounded away of zero. Suppose this were not the case; Then there is a sequenceqn ∈ Γ, with kΓ(qn) → 0. Translate M horizontally so that qn transforms to afixed point σ. The curvature bounds allows us to take a limit surface M∞ in thewhole R3 (not only in the half space x3 > 0) of the translated surfaces. Thissurface M∞ is not necessarily embedded (it could have tangential selfintersectionsat the level x3 = 0) but retains any other properties of surfaces in S. In particular,assertion (ii) applies to M∞. Moreover its width is smaller than 1/H and thefunction ψ∞ constructed as in (7) on (M∞)+ is nonpositive (because it is alimit on nonpositive functions) and vanishes at σ. As the curvature of the curveM∞ ∩ P at σ is zero, we conclude from (ii) that one of the components of M∞must be a cylinder. This contradiction proves that kΓ < 1/ρ for some positiveconstant ρ. This implies that any connected component of Γ is a closed Jordancurve contained in a disk of radius ρ. Therefore, if the number of componentsof P−Ω where finite, then Ω would contain arbitrarily large round disks, whichis clearly impossible (use for instance Theorem 2). This completes the proof ofthe theorem.

DEPARTAMENTO DE GEOMETRIA Y TOPOLOGIA, FACULTAD DE CIENCIAS, UNIVERSIDAD

DE GRANADA, 18071 GRANADA, SPAIN

E-mail: [email protected]

INSTITUTO DE MATHEMATICA PURA E APLICADA (IMPA), ESTRADA DONA CASTORINA

110, 22460-320 RIO DE JANEIRO RJ, BRAZIL

E-mail: [email protected]


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