j. differential geometry 57 (2001) 493-508

HALF-SPACE THEOREMS FOR MINIMAL SURFACES WITH BOUNDED CURVATURE

G. PACELLI BESSA, LUQUÉSIO P. JORGE & G. OLIVEIRA-FILHO

Abstract

First we prove a version of the Strong Half-Space Theorem for minimal surfaces with bounded curvature in ℝ^{3}. With the techniques developed in our proof we give criteria for deciding if a complete minimal surface is proper. We prove a mixed version of the Strong Half-Space Theorem. Turning to 3-dimensional manifolds of bounded geometry and positive Ricci curvature, we show that complete injectively immersed minimal surfaces with bounded curvature are proper and as a corollary we have a Half-Space Theorem in this setting. Finally we show an application of the maximum principle for nonproper minimal immersions in ℝ^{3}.

1. Introduction

The Strong Half-Space Theorem [8], states that two complete, minimally and properly immersed surfaces in ℝ^{3} intersect unless they are parallel planes. The word strong there stands in opposition to weak in the version where one of the surfaces is a plane and will be refered as Half-Space Theorem.

There is an extension, due to Anderson and Rodriguez [2], of the Strong Half-Space Theorem that shows that in a complete oriented noncompact 3-dimensional Riemannian manifold N with nonnegative Ricci curvature Ric_{N} ≥ 0 and sectional curvature bounded from above K_{N} ≤ b, any two complete properly immersed oriented minimal surfaces, intersect unless they are totally geodesic and parallel leaves in a local product structure. In all these results, properness is required.

On the other hand, Xavier [16] proved a version of the (weak) Half-Space Theorem where instead of properly immersed, he required bounded curvature. He showed that the convex hull of a complete nonplanar minimal surface with bounded curvature in ℝ^{3} is all of ℝ^{3}. In this paper, we prove an extension of Xavier’s Half-Space Theorem in the same way that the Strong Half-Space extends the weak Half-Space Theorem. We prove the following theorem:

Received April 2, 2001.

Theorem 1.1. Let M_{1} and M_{2} be complete minimal immersed surfaces in ℝ^{3} with bounded curvature. Then M_{1} ∩ M_{2} ≠ ∅ unless they are parallel planes.

This result raises the problem of whether the hypothesis of properly immersed in minimal surface theory can be replaced by geometric hypotheses. One may take two points of view in looking at this problem. First, one can try to prove theorems that substitute geometric hypotheses for the hypothesis of proper immersion. Theorem 1.1 fits this point of view. Second, one can look for geometric hypotheses that imply that a complete minimal surface is proper. We should remark that, requiring bounded curvature for a complete, minimally immersed surface, does not guarantee its properness. For instance, in [3] Andrade constructs a complete immersion of the plane ℂ into ℝ^{3} with bounded curvature, dense in a proper and unbounded subset with nonempty interior of ℝ^{3}. See also [12] for another example.

The ideas developed in the proof of Theorem 1.1 can be applied to prove theorems in the spirit of the second point of view, about complete minimal surfaces with boundary and bounded curvature. Here, for a complete surface with boundary, we understand a surface with boundary where all Cauchy sequences converge. We prove the following theorem:

Theorem 1.2. Let φ : M ↪ ℝ^{3} be a complete minimally immersed surface with boundary ∂M (possibly empty) and bounded curvature such that φ(M) ⊂ \overline{Ω}, where Ω is a mean convex domain. If ∂M ≠ ∅, we suppose that φ |_{∂M}: ∂M ↪ ∂Ω ⊂ ℝ^{3} is proper. Then one of the following conditions holds:

i) φ is proper.

ii) The limit set Lim φ, (see Definition 2.2), is a union of parallel planes lying in the interior of a slab or in a half-space inside \overline{Ω}.

In both cases, there are planes separating ∂ Ω from Lim φ, unless ∂Ω is a plane contained in the limit set.

As a corollary of Theorem (1.2) we have the following criteria for deciding whether a complete minimal surface of bounded curvature is proper.

Corollary 1.3. Let Σ be a complete nonflat minimal surface in ℝ^{3}. Let φ : M ↪ ℝ^{3} be a complete minimal immersion with bounded curvature transversal to Σ. Set Γ = φ^{−1}(Σ). Assume that φ |_{Γ}: Γ ↪ ℝ^{3} is proper. Suppose that one of the following conditions holds:

i) Σ is proper.

ii) Σ has bounded curvature.

Then φ is proper.

In particular, we have the following Mixed Half-Space Theorem.

Corollary 1.4 (Mixed Half-Space Theorem). Let M_{1} be a complete proper minimal surface and M_{2} be a complete minimal surface with bounded curvature in ℝ^{3}. Then M_{1} ∩ M_{2} ≠ ∅, unless they are parallel planes.

We now turn to complete 3-dimensional Riemannian manifolds with bounded geometry and positive Ricci curvature. Bounded geometry here means sectional curvature bounded from above and injectivity radius bounded away from zero.

Theorem 1.5. Let φ : M^{n} ↪ N^{n+1} be a complete minimal immersed hypersurface with scalar curvature bounded from below in a complete dimensional Riemannian manifold N of bounded geometry. Suppose in addition that N has nonnegative Ricci curvature Ric_{N} ≥ 0. Then φ is proper or every orientable leaf S ⊂ Lim φ such that S ∩ φ(M) = ∅ is stable. Moreover, if S is compact then S is totally geodesic and the Ricci curvature of N is identically zero in the normal directions to S.

Corollary 1.6. Let M ⊂ N be a complete oriented injectively and minimally immersed surface with bounded curvature in a 3-dimensional Riemannian manifold of bounded geometry and positive Ricci curvature. Then:

1. If N is compact then M is compact.

2. If N is not compact then M is proper.

Corollary (1.6) together with the Anderson-Rodriguez Half-Space Theorem yield the following theorem.

Theorem 1.7. Let M_{1} and M_{2} be complete, minimally and injectively immersed surfaces with bounded sectional curvature in a complete, noncompact 3-dimensional Riemannian manifold of bounded geometry and positive Ricci curvature. Then M_{1} ∩ M_{2} ≠ ∅, unless they are both totally geodesic and parallel leaves in a local product structure.

Finally we present an alternate proof of a result that is a direct corollary of Theorem (1.1) because it shows an application of the maximum principle for nonproper minimal immersions.

Let M be a complete minimal surface of ℝ^{3} with bounded curvature and let C be a catenoid. Then M ∩ C ≠ ∅.

H. Rosenberg independently has proven Theorem 1.1 and Corollary 1.6, (1) (see [13]). In proving a Half-Space Theorem in ℝ^{3}, one follows the same idea as to prove Hoffman-Meeks Strong Half-Space Theorem, i.e., one needs to construct a complete and stable minimal surface separating M_{1} and M_{2}. For this, one constructs mean convex barriers, and our proof differs from Rosenberg’s in the way these barriers are constructed. Theorem 1.1 and Corollary 1.6 were divulged by the second author in the regular seminar at IMPA-Rio de Janeiro in November 1998.

2. Half-Space Theorem (Theorem 1.1)

This section is divided in two subsections. In the first subsection, we construct a mean convex barrier and in the second one we present the proof of Theorem 1.1.

2.1 Barrier construction

Throughout this section we denote by B_{R} an open Euclidean ball with radius R centered at the origin and when it has another center p we write B(p, R) instead. For a set A we will denote by A its closure in ℝ^{3} and by T_{ϵ}(A) = {p ∈ ℝ^{3} ; dist(p, A) ≤ ϵ} its closed ϵ-tubular neighborhood. Let us consider M ⊂ ℝ^{3} a complete nonproper minimal surface with bounded Gaussian curvature such that M ≠ ℝ^{3}. Set r_{0} < (sup_{x∈M} \sqrt{|k_{M}(x)|})^{−1}, where k_{M}(x) is the Gaussian curvature of M at x in such a way that T_{r_{0}}(M) is not all of ℝ^{3}. To construct the barrier we will need some results from [7] about sets with positive reach and the catograph set of a function . The reach of a subset A ⊂ ℝ^{3} is the largest ϵ (possibly ∞) such that if x ∈ ℝ^{3} and the distance d(x, A) is smaller than ϵ, then A contains a unique point x nearest to x. If f : A → ℝ is a function, then the catograph set of f is defined as U = {(x, y) ∈ A × ℝ; 0 ≤ y ≤ f(x)}.

Lemma 2.1. The tubular neighborhood T_{r}(M), for a.a. 0 < r ≤ r_{0}, has Lipschitz boundary and its complement \overline{ℝ^{3} \ T_{r}(M)} has positive reach. Also every boundary point of T_{r}(M) has inner support spheres of radius t, for 0 < t ≤ r. In particular, the outside tangent cone of ∂ T_{r}(M) has no angle bigger than π.

Proof. That the tubular neighborhood T_{r}(X) has Lipschitz boundary and its complement \overline{ℝ^{3} \ T_{r}(X)} has positive reach are proved in [7] (see Main Theorem) by J. Howland Fu for X bounded. Taking X = M ∩ B_{R} and making R going to infinity we conclude that ∂ T_{r}(M) is Lipschitz.

For each p ∈ ∂ T_{r}(M) there exists at least a point p_{0} ∈ M such that dist(p, M) = |p−p_{0}|. Observe that p ∈ ∂ B(p_{0}, r) and B(p_{0}, r) ⊂ T_{r}(M). Thus ∂ B(p_{0}, r) is a support at p for ∂ T_{r}(M) inside T_{r}(M). For any point p′ in the interior of the line segment [p_{0}, p], the ball B(p′, |p′ − p|) touches ∂ T_{r}(M) at exactly one point. Moving p on ∂ T_{r}(M) keeping |p′ − p| constant, we get that the reach of \overline{ℝ^{3} \ T_{r}(M)} is no less than |p′ − p| or even better, than r.

Now assuming that the tangent cone of ∂ T_{r}(M) is not a plane, we may have more than one point p_{0} ∈ M realizing the distance to p. But for each of such points we get a support sphere for ∂ T_{r}(M) at p inside T_{r}(M). In particular, the tangent cone is inside the cone determined by the intersection of these spheres at p. Therefore no plane section has angle bigger than π.

We will need a basic lemma (Lemma 2.3) about limit sets of isometric immersions φ : M^{m} ↪ N^{n}, developed in [4] in more general situation than the one considered here. For completeness, we first give the definition of limit sets.

Definition 2.2. Let φ : M^{m} ↪ N^{n}, 1 ≤ m < n, be an isometric immersion where M and N are complete Riemannian manifolds of dimensions m and n, respectively. The limit set of φ, denoted Lim φ, is the following set:

Lim φ = {p ∈ N ; ∃ {p_{n}} ⊂ M, dist_{M}(p_{0}, p_{n}) → ∞
and dist_{N}(p, φ(p_{n})) → 0}

= ⋂_{K⊂M} \overline{φ(M) \ φ(K)}, K compact.

Observe that Lim φ ⊂ φ(M) is a closed set and Lim φ = ∅ if and only if φ is proper. Sometimes when the immersion is not explicitly presented (i.e., M ⊂ N) we denote the limit set by Lim M. We have the following lemma.

Lemma 2.3. Let M ⊂ ℝ^{3} be a complete nonproper minimal surface with bounded curvature and let p ∈ Lim M. Then there exists a sequence of minimal disks D_{i} ⊂ M converging uniformly (in the C^{∞}-topology) to a minimal disk D ⊂ Lim M containing p. Moreover, the limit disk D can be extended to a complete minimal surface S_{p} ⊂ Lim M passing through p with bounded curvature. If the limit disk D is flat, then S_{p} is a plane.

This lemma is a consequence of the fact that each point x ∈ M has a neighborhood V_{x} that can be graphed over a ball in the tangent plane of M at x with radius uniformly bounded from below, coupled with convergence results of minimal graphs.

Lemma 2.4. Let M ⊂ ℝ^{3} be a complete nonproper minimal surface with bounded curvature. Let us assume that no limit disk given by Lemma (2.3) is flat. Given p ∈ ∂ T_{r}(M), 0 < r < r_{0}, then there is a catograph set U_{p} of some function f such that:

(i) p ∈ Int(U_{p}).

(ii) S_{p} = (∂ U_{p}) \ T_{r}(M) is a compact embedded surface with nonnegative mean curvature for unit normal vector pointing outside U_{p} and ∂S_{p} ⊂ ∂ T_{r}(M).

(iii) For p, q ∈ ∂ T_{r}(M) with the corresponding surfaces S_{p} and S_{q} intersecting in p′ interior to both surfaces, the outside angle of U_{p} ∩ U_{q} at p′ is at most π.

Proof. Let p ∈ ∂ T_{r}(M) and p_{0} ∈ M as before. There exists a sequence of minimal embedded disks D_{k} ⊂ M converging uniformly to a minimal embedded disk D_{0} ⊂ M containing p_{0}. It is clear that D_{0} ∩ B(p, r) = ∅, otherwise there would exist a point q ∈ D_{0} such that dist(p, q) < r contradicting the fact that p ∈ ∂ T_{r}(M). Now let ν be a continuous unit normal vector field on D_{0} so that ν(p_{0}) is pointing toward p. We may assume that the image of D_{0} by ν takes the value ν(p_{0}) (in the unit sphere) only once with multiplicity. Otherwise, D_{0} would be a flat disk. The parallel disks D_{t} = {x(t) := x+tν(x); x ∈ D_{0}}, t ≤ r, are well defined and embedded provided that r ≤ r_{0}. Its mean curvature H_{t} is given by

H_{t}(x(t)) = \frac{−k_{M}(x) t}{1 + k_{M}(x) t^{2}} ≥ 0, t ≥ 0. (1)

The line segment [p_{0}, p] = {p_{0} + tν(p_{0}), 0 ≤ t ≤ r} is perpendicular to ∂ B(p_{0}, r) at p and to the tangent space T_{p_{0}}D_{0}. Let C be the solid cylinder with axis generated by ν(p_{0}) and orthogonal cross section the disk B = B(p_{0}, ϵ) ∩ T_{p_{0}}D_{0}. We may assume that ∂D_{t}, (0 < t ≤ r) is outside C and D_{t} ∩ C is a graph over B. The surface D_{r} is also a support for ∂ T_{r}(M) at p.

Fix t_{0}, 0 < t_{0} < r and observe that D_{t_{0}} + (r − t_{0})ν(p_{0}) is contained in T_{r}(M) with the boundary in the interior. To see this take q ∈ D_{t_{0}} ∩C and q′ ∈ D_{0} the nearest point to q. Then |q′ − (q + (r − t_{0})ν(p_{0}))| < |q′ − q| + r − t_{0} = r. The inequality is strict because (|q′ − q| = t_{0}) and ν(q′) ≠ ν(p_{0}). Hence, D_{t_{0}} + (r − t_{0})ν(p_{0}) lies in one side of D_{r} and touches D_{r} at p. We can choose δ > 0 so that:

(i) The disk D_{t_{0}} ∩C +(δ +r −t_{0})ν(p_{0}) crosses the boundary ∂ T_{r}(M), dividing it in at leas t two sets, one of them being a small disk with p in the interior.

(ii) The boundary of D_{t_{0}} ∩ C + (δ + r − t_{0})ν(p_{0}) is contained in the interior of T_{r}(M).

Writing D_{t_{0}} ∩C +(δ+r−t_{0})ν(p_{0}) as graph of a function f over B, we set U_{p} as the catograph set of f, that is, U_{p} = {(x, y) ∈ C; 0 ≤ y ≤ f(x)}. The assertion (iii) follows from this construction.

Conclusion 2.5. Let ℝ^{3} \ M ≠ ∅ and M ∩ B_{R} ≠ ∅. Let X ⊂ B_{R} be a closed set not intersecting M. Then there is an ϵ = ϵ(R, M) and a C^{∞} by parts surface S_{ϵ} such that:

(i) S_{ϵ} ⊂ T_{2ϵ}(M) \ T_{ϵ}(M).

(ii) S_{ϵ} ∩ T_{2ϵ}(X) = ∅.

(iii) S_{ϵ} is part of the boundary isolating M and T_{2ϵ}(X) and is mean convex with respect to the open set between them.

Remark 2.6. These surfaces S_{ϵ} can be made by minimal disks. Just consider t_{0} = 0 in their construction as above. One has to show that in that case we still have D_{0} + rν(p_{0}) contained in T_{r}(M) with the boundary in the interior. But this is true, for if we take a point p_{0} ≠ q ∈ D_{0} ∩ C, the distance between q + rν(p_{0}) and D_{0} is r if and only if ν(q) = ν(p_{0}). The rest of the construction is the same.

Remark 2.7. Conclusion 2.5 is true if we replace M by Lim M.

2.2 Proof of Theorem 1.1

Let M_{i} ⊂ ℝ^{3}, i = 1, 2 be two complete nonproper minimal surfaces with bounded Gaussian curvature.

1^{st}. Assume that M_{1} ∩ M_{2} ≠ ∅ and M_{1} ∩ M_{2} = ∅. Take a point p ∈ M_{1} ∩ M_{2}. There are two sequences of minimal disks D_{k}^{i} ⊂ M_{i} i = 1, 2 converging uniformly to minimal disks D^{i} ⊂ M_{i}, i = 1, 2, both containing p. D^{1} and D^{2} can be extended to complete minimal surfaces with bounded curvature S_{p}^{1} ⊂ M_{1} and S_{p}^{2} ⊂ M_{2} respectively. They can not intersect themselves transversally, otherwise M_{1} would intersect M_{2}. Therefore they are tangent to each other at p and lie to one side of the common tangent plane at p. Then by the maximum principle S_{p}^{1} = S_{p}^{2} := S_{p}. Clearly S_{p} does not intersect M_{i}, i = 1, 2, nor has self-intersections because we would have M_{1} ∩ M_{2} ≠ ∅. By Theorem 1.5 below, S is stable and by [5] or [6], S is a plane. S separates M_{1} from M_{2} or they are at the same side of S. In both cases, by Xavier’s Half-Space Theorem, M_{1} and M_{2} are parallel planes. This contradicts the nonproperness assumption.

2^{nd}. Suppose that M_{1} ∩ M_{2} = ∅ and M_{1}, contains a flat limit disk D. Then D can be extended to a plane S. By Xavier’ s Half-Space Theorem M_{2} intesects S and thus M_{1}, unless M_{2} is a plane parallel to S. This contradicts the hypotheses that M_{2} is nonproper and M_{2} does not intersect M_{1}.

3^{rd}. Assume now that M_{1} ∩ M_{2} = ∅ and M_{i}, i = 1, 2 has no flat limit disk. Now let R_{0} > 0 be such that B_{R_{0}} intersects M_{i}. Taking X as M_{i} ∩ B_{R_{0}}, for i = 1, 2, we find ϵ_{0} = ϵ_{0}(M_{i}, R_{0}) such that the surfaces S_{ϵ i} of Conclusion 2.5 together with parts of ∂ B_{R_{0}} are the boundary of a mean convex set Ω(R_{0}) isolating T_{ϵ_{0}}(M_{1}) and T_{ϵ_{0}}(M_{2}) inside B_{R_{0}}. Let x_{1} ∈ M_{1} ∩B_{R_{0}} and x_{2} ∈ M_{2} ∩B_{R_{0}} realize the distance of these two sets. The line segment [x_{1}, x_{2}] intersects M_{1} and M_{2} only at the end points. Consider Ω(R_{0}) with the usual orientation and [x_{1}, x_{2}] oriented from x_{1} to x_{2}. Let S be the connected component of S_{ϵ 1} crossing [x_{1}, x_{2}]. By construction the intersection number of this set is exactly one.

Suppose there is a closed curve γ ⊂ S homotopic to a point in Ω(R_{0}). By [10] (or [15]), one has that γ bounds an embedded minimal disk D_{γ} inside Ω(R_{0}). Cut S and Ω(R_{0}) along D_{γ} and glue two copies of D_{γ} along the boundary. This surgery does not change the intersection number with [x_{1}, x_{2}]. It also produces a new mean convex domain. Doing this a finite number of times we get an incompressible surface S(R_{0}) in a new mean convex domain Ω̃(R_{0}) crossing [x_{1}, x_{2}]. We claim that S(R_{0}) has boundary and, of course ∂ S(R_{0}) ⊂ ∂ B_{R_{0}}. If not, S(R_{0}) bounds M_{1} or M_{2} once they are on opposite sides. But there are no complete bounded minimal surfaces in ℝ^{3} with bounded Gaussian curvature (see [9]). Hence using again [10] or [15] we get an stable minimal surface S_{0} ⊂ B_{R_{0}} such that:

(i) ∂ S_{0} = ∂ S(R_{0}) in ∂ B_{R_{0}}.

(ii) S_{0} is homotopic to S(R_{0}) in Ω̃(R_{0}).

(iii) The intersection number of S_{0} and [x_{1}, x_{2}] is one.

(iv) S_{0} separates points of M_{1} and M_{2} inside B_{R_{0}}.

Take a divergent increasing sequence of R_{j} and corresponding S_{j} ⊂ B_{R_{j}} stable minimal surfaces satisfying (i)–(iv). It is well known that S_{j} converges in compact parts to a complete stable minimal surface S immersed in ℝ^{3} (see [1]), and separating points of M_{1} and M_{2}, (x_{1} and x_{2}, for instance). By [5] or [6], S is a plane and M_{1} and M_{2} are on opposite sides of this plane S. By Xavier’s Half-Space Theorem [16], M_{1} and M_{2} are parallel planes, contradicting the nonproperness assumption.

To finish the proof of Theorem (1.1) we need to consider the case that one of the surfaces is proper with bounded curvature. This is done in the next section in a more general context, i.e., one of the surfaces is proper regardless the bounds on the curvature.

3. Properness criteria

Here we present the proof of Theorem 1.2 and its corollaries. Let Ω be a mean convex domain with boundary Σ = ∂ Ω. By definition of mean convex sets, Σ is a union of pieces of regular surfaces with nonnegative mean curvature with respect to normal vector field pointing toward the interior of Ω, glued by their boundaries with inner angle less than or equal to π. If a proper minimal surface M inside Ω touches one face Σ′ at an interior point x_{0}, then by the maximum principle Σ′ is contained in M. If x_{0} ∈ ∂ Σ′, then x_{0} is also in the boundary of a neighbor face Σ′′ and these faces are tangent to M. In a similar way, (by the maximum principle at the boundary), we can conclude that Σ′ and Σ′′ are contained in M, if M is large enough. Moreover, if Σ has a compact component, then Ω is compact. Further, if there is a plane ℙ_{0} ⊂ Ω and if we let ℙ′_{0} ⊂ Ω be a plane parallel to ℙ_{0} closest to Σ, then ℙ′_{0} ∩ Σ ≠ ∅, implies that ℙ′_{0} ⊂ Σ.

3.1 Proof of Theorem 1.2

Now suppose that Ω is a mean convex domain with boundary Σ = ∂ Ω and φ : M ↪ ℝ^{3} is a complete nonproper minimally immersed surface with boundary and bounded curvature such that φ(M) ⊂ \overline{Ω}. Assume that if ∂M ≠ ∅, then φ |_{∂M}: ∂M ↪ ∂ Ω is proper. Hence, there is no divergent sequence {x_{k}} in M with {φ(x_{k})} accumulating in Σ. It follows that the limit set Lim φ exists and it is a union of complete immersed minimal surfaces with bounded curvature inside Ω.

Lemma 3.1. Under the above condition, there exists a plane ℙ_{0} separating Lim φ from ∂ Ω, unless ∂Ω is a plane ℙ_{0} ⊂ Lim φ. Moreover all S_{p} in Lim φ are planes.

Proof. If Lim φ ∩ ∂Ω ≠ ∅ then ∂Ω is a leaf from Lim φ. By a small modification of Theorem 1.5 we have that ∂Ω is stable and therefore a plane. Otherwise, take a ball B_{R} intersecting Lim φ and ∂Ω. Choose a point x_{1} ∈ Lim φ and x_{2} ∈ ∂Ω nearest to x_{1}. Without loss of generality, we may assume that the line segment L = [x_{1}, x_{2}] from x_{1} to x_{2} lies inside Ω and does not intersect Lim φ and ∂Ω at interior points of L. Following the proof of Theorem 1.1, we can construct a plane ℙ_{0} in Ω intersecting L and separating Lim φ from ∂ Ω. Since S_{p}, for p ∈ Lim φ, has bounded curvature, by Xavier’s Half-Space Theorem S_{p} must be a plane parallel to ℙ_{0}. Therefore, for each point p ∈ Lim φ, S_{p} ⊂ Lim φ is a plane parallel to ℙ_{0}.

Lemma 3.1 finishes the proof of Theorem 1.2.

3.2 Corollaries 1.3 & 1.4

Suppose that φ : M ↪ ℝ^{3} is complete nonproper minimal immersion with bounded Gaussian curvature and Γ = φ^{−1}(Σ) is proper.

Case i). ℝ^{3} \ Σ is a union of mean convex domains. Choose a complete noncompact connected component M′ of M \ Γ. Suppose that φ |_{M′}: M′ ↪ ℝ^{3} is not proper. Then by Theorem 1.2, there is a plane separating the limit set of φ |_{M′} from Σ. By Xavier’s Half-Space Theorem Σ is a plane, a contradiction to the hypothesis. Thus the restriction of φ to any noncompact component M′ of M \ Γ is proper and φ : M ↪ ℝ^{3} itself is proper.

Case ii). Suppose that φ is not proper. Observe that φ(M) ∩ \overline{Σ} = φ(Γ). For, if there is a point x ∈ \overline{Σ} ∩ φ(M), then there is a sequence of disks D_{k} ⊂ Σ converging to a disk D ⊂ \overline{Σ} containing x. These disks D_{k} intersect φ(M) in a sequence of points of φ(Γ) converging to x. Since φ(Γ) is closed, (φ_{|Γ} is proper), x is in φ(Γ). φ(Γ) ⊂ φ(M) ∩ \overline{Σ} is obvious. If Lim φ ≠ ∅, by Theorem (1.1) Lim φ intersects Σ. By the same reasoning as above, Lim φ ∩ Σ ⊂ φ(Γ) and φ_{| Γ} is not proper, a contradiction.

Proof of Corollary (1.4). Set Σ = M_{1}. If Γ = M_{1} ∩ M_{2} = ∅ then Γ is proper and M_{2} is also proper by Case i) of Corollary (1.3). By the Strong Half-Space Theorem they are parallel planes.

4. Proof of Theorem 1.5

Let φ : M ↪ N be a complete minimal hypersurface with scalar curvature bounded from below in a complete Riemannian manifold N of bounded geometry. Recall (Lemma 2.3) that for each p ∈ Lim φ there exists a complete minimal hypersurface S ⊂ Lim φ with scalar curvature bounded from below passing through p, and a sequence p_{k} ∈ φ(M) converging (up to a subsequence) to p, moreover for each compact set C_{p} ⊂ S containing p there is a sequence of compacts C_{k} ⊂ φ(M) containing p_{k} converging uniformly to C_{p}. Such a hypersurface S is called a leaf passing through p. In this section we shall prove Theorem 1.5.

Theorem 1.5. Let φ : M^{n} ↪ N^{n+1} be a complete minimal immersed hypersurface with scalar curvature bounded from below in a complete dimensional Riemannian manifold N of bounded geometry. Suppose in addition that N has nonnegative Ricci curvature Ric_{N} ≥ 0. Then φ is proper or every orientable leaf S ⊂ Lim φ such that S ∩ φ(M) = ∅ is stable. Moreover, if S is compact then S is totally geodesic and the Ricci curvature of N is identically zero in the normal directions to S.

Remark 4.1. For n = 2, Schoen shows in [14] that S is totally geodesic if it is noncompact. The proof of this result is close to parts of proofs done by Fisher-Colbrie-Schoen [6]. We will include it here for the sake of completeness.

Proof. Suppose that Lim φ ≠ ∅, that is φ is not proper. Let S ⊂ Lim φ be an orientable leaf such that S ∩ φ(M) = ∅ hence S has no self intersections. Let C ⊂ S be a compact and proper subset of S and T_{ϵ}(C) an embedded ϵ-tubular neighborhood of C in N. There exists a sequence of compact sets C_{k} ⊂ φ(M) converging uniformly to C. We may assume that for k ≥ k_{0} the sets C_{k} are injectively immersed and C_{k} ⊂ T_{ϵ}(C). Passing to a subsequence if necessary we may assume that {C_{k}} converges to C by one side of C. Let U_{ϵ} be this side and ν be a continuous unit normal (to S) vector field on S pointing towards U_{ϵ}. Now (following Ros [11]) we let L be the Jacobi operator on S, i.e., L = Δ + Ric(ν) + |A|^{2}, where Δ is the Laplacian of S, Ric(ν) is the Ricci curvature of N in the direction ν and |A| is the norm of the second fundamental form of S ⊂ N. Take a larger compact set C′ containing C properly and consider a converging sequence of compact sets also denoted by C_{k} converging to C′. We may assume that the only solution of L v = 0 on C′ and v = 0 on ∂ C′ is the function v ≡ 0. Therefore, there exists a function u ∈ C^{∞}(C′) such that L u = 1 on C′ and u = 0 on ∂ C′. The mean curvature H(t), |t| < ϵ, of the immersions ψ_{t} : C′ → ℝ^{3}, ψ_{t}(x) = exp_{x}(t u(x) ν(x)) for all x ∈ C′, has derivative at t = 0 given by 2H′(0) = L u = 1 on C′. Thus if ϵ is small, H(t) > 0 on C′, 0 < t < ϵ. If u is positive at some interior point of C then ψ_{t}(C′) , t < ϵ, has a tangency point with some C_{k} and this is not allowed by the maximum principle, since C_{k} is a minimal surface and the mean curvature is positive with respect to the vector pointing to that direction. So u ≤ 0. If u(q) = 0, q ∈ int(C′), by the same reasons as above we have that u ≡ 0, and this is impossible. Thus u < 0 in the interior of C′ and u = 0 on the ∂C′. Setting w = −u we have that w is a positive function on the interior of C′ and L w ≤ 0. In particular w restricted to the boundary of C is a positive function. Let u_{1} be the first eigenfunction of C, i.e., L u_{1} = −λ_{1}(C)u_{1}. Suppose by contradiction that λ_{1}(C) < 0. Let h = w − t u_{1} > 0 for some small t > 0. We have that L h = L w − tL u ≤ 0 + tλ_{1} u_{1} < 0 on C. Then Δ h < 0 on C and h has a minimum in the interior. By the maximum principle h is constant. Choosing t in a way that this minimum is zero we have a contradiction. This shows that λ_{1}(C) is positive and thus C is stable.

If S is not compact then there exists an exhaustion of S by compact sets, each one stable. Thus S is stable. When S is compact we will need the following theorem due to Fisher-Colbrie-Schoen [6].

Theorem 4.2 (Fisher-Colbrie-Schoen). Let (M, ds^{2}) be a closed Riemannian manifold and let q be a smooth function on M. Given any proper domain D in M, let λ_{1}(D) < λ_{2}(D) ≤ λ_{3}(D) ≤ … be the sequence of eigenvalues of Δ − q acting on functions that vanish on ∂D. If λ_{1}(D) > 0 for all proper domains D, then there exists a positive function g satisfying the equation Δg − q g = 0 in M.

This theorem is a part of Theorem 1 of [6] that is valid for closed Riemannian manifolds. Every proper compact domain C ⊂ S is stable and then the first eigenvalue λ_{1}(C) is positive for the stability operator Δ + Ric ν + |A|^{2} = Δ − q. Here A is the second fundamental form of S ⊂ N and ν is a unit vector field in S and normal to S; thus by Theorem 4.2, there exists a positive function g in S satisfying Δg−q g = 0. Therefore,

∫_{S} Δg − ∫_{S} q g = 0 ⇒ ∫_{S} q g = 0 ⇒ q = 0 ⇒ |A| = 0 and Ric ν = 0.

Then S is totally geodesic and the Ricci curvature is zero in the normal directions to S.

The stability operator is then L = Δ, acting on functions f : S → ℝ with ∫_{S} f = 0. Now suppose that S is not stable. Then λ_{1}(S) < 0 and Δf + λ_{1}(S)f = 0 in S for some function f : S → ℝ with ∫_{S} f = 0. Let D_{f} = {x ∈ S : f(x) > 0} be the nodal set of f. Thus

λ_{1}(D_{f}) = inf{ ∫_{D_{f}} uΔu/ ∫ u^{2}; supp u ⊂ D_{f}} ≤ ∫_{D_{f}} fΔf/ ∫ f^{2}
= − ∫_{D_{f}} |∇f|^{2}/ ∫ f^{2} < 0.

This contradicts the fact that λ_{1}(C) > 0 (stability of compact proper subsets) of any compact subset. Therefore S is stable. For n = 2, Schoen [14] has shown that a complete (non compact) stable minimal surface in a 3-dimensional Riemannian manifold with nonnegative Ricci curvature is totally geodesic.

Now we can prove Corollary 1.6 as follows. Suppose that φ : M ↪ N is a complete noncompact minimally and injectively immersed surface M with sectional curvature bounded from below into a 3-dimensional compact Riemannian manifold N with positive Ricci curvature. Since M is not compact, Lim φ ≠ ∅. Let S ⊂ Lim(S). By hypotheses, φ(M) has no self intersections, hence S has no self intersections either and φ(M) ∩ S = ∅. Since Ric_{N} > 0 then the first Betti number of N, b_{1}(N) = 0. Thus S is orientable. By Theorem 1.5 S is stable and totally geodesic. By Corollary 3 of [14] S is compact; in fact S is conformally equivalent to the sphere 𝕊^{2}. Again, by Theorem 1.5, the Ricci curvature Ric_{N}(ν) ≡ 0 in the normal directions to S, a contradiction.

Suppose that N is not compact and M is not proper, then Lim φ ≠ ∅. There is a leaf S ⊂ Lim φ such that S ∩ φ(M) = ∅ since M is injectively immersed. By Corollary 3 of [14], S is compact and thus the Ricci curvature in the normal directions are zero, contradiction. Therefore, Lim φ = ∅ and M is proper.

5. An application of the maximum principle

In this section we present a particular case of Theorem 1.1 to show a way that the maximum principle can be applied to nonproper minimal immersions.

Corollary 5.1. Let φ : M ↪ ℝ^{3} be a complete minimal immersed surface in ℝ^{3} with bounded sectional curvature and let C be any catenoid in ℝ^{3}. Then M ∩ C ≠ ∅.

Suppose by contradiction that M ∩ C = ∅. We will assume that Lim φ ≠ ∅ (otherwise the Strong Half-Space Theorem implies the claim) and it is connected. Observe that Lim φ ≠ C because C is not stable (see Theorem 1.5). In fact, we may suppose that Lim φ ∩ C = ∅, because otherwise it would imply that M intersects C. So we have that M neither intersect C nor accumulates on C. Let Ω(C) be the simply connected open region of ℝ^{3} whose boundary is C. We may assume that C does not intersects the x_{3}-axis (after a rotation of ℝ^{3}). Let B_{R} be a closed ball in ℝ^{3} centered at the origin and radius R such that B_{R} ∩ Lim φ ≠ ∅. Suppose first that Lim φ ⊂ Ω(C). If Lim φ does not intersect the plane x_{3} = 0 there is a point q ∈ B_{R} ∩ Lim φ closest to the plane {x_{3} = 0} with positive distance. Move the plane {x_{3} = 0} parallely, (i.e the planes are {x_{3} = t}) towards q till it touches the first point p (possibly q) in the compact set B_{R} ∩ Lim φ say, at x_{3} = t_{0}. By Theorem 1.5 there is a complete minimal surface S ⊂ Lim φ passing through p. This minimal surface S touches the plane {x_{3} = t_{0}} at p but does not cross it because p is the closest point in B_{R} ∩ Lim φ to the plane {x_{3} = 0} and a piece of S is still in the compact B_{R} ∩ Lim φ. By the maximum principle, S is the plane {x_{3} = t_{0}} and it must intersect C. So, if Lim φ ⊂ Ω(C), then it does intersect the plane {x_{3} = 0}, in fact the reasoning above shows that it intersects all the planes {x_{3} = t}. Recalling that B_{R}∩Lim φ is compact and does not touch B_{R}∩C, we then make a homothety of C, shrinking the catenoid C to another catenoid C̃ till it touches a first point p̃ ∈ B_{R} ∩Lim φ. With the same reasoning the maximum principle applies and we have that Lim φ∩C ≠ ∅. Therefore, Lim φ ⊂ [ℝ^{3} \ \overline{Ω}(C)]. In this case there is a point q̂ ∈ B_{R} ∩ Lim φ closest to the x_{3}-axis. We make a homothety of C to enlarge it to another catenoid Ĉ that first touches at a point p̂ ∈ B_{R} ∩Lim φ. Again, the minimal surface S(p̂) would coincide to Ĉ and would intersect C. In any of the cases we have a contradiction to the hypotheses Lim φ ≠ ∅ and Lim φ ∩ C = ∅.

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