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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


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Volume 19, Issue 1

Issues

Minimal hypersurfaces in ℝn × Sm

Jimmy Petean / Juan Miguel Ruiz
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/advgeom-2017-0060

Abstract

We classify minimal hypersurfaces in ℝn × Sm with n, m ≥ 2 which are invariant by the canonical action of O(n) × O(m). We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvature hypersurfaces are all unstable.

Keywords: Minimal hypersurface

MSC 2010: 53C42

1 Introduction

In this article we construct families of complete embedded minimal hypersurfaces in Riemannian products ℝn × Sm, and also some examples of families of complete embedded hypersurfaces of constant mean curvature. The study of minimal hypersurfaces is a very classical problem in differential geometry; there is a large literature on construction of examples. The most studied cases are the minimal surfaces in S3 and ℝ3. Let us mention for instance that Lawson proved in [7] that every compact orientable surface can be embedded as a minimal surface in S3. Further constructions of minimal surfaces in the sphere were done by Karcher, Pinkall and Sterling in [6] and by Kapouleas and Yang in [5], among others. There are also plenty of constructions in other 3-manifolds, for instance recently Torralbo in [17] built examples of minimal surfaces immersed in the Berger spheres. And in the last few years there has been great interest in the study of minimal surfaces in 3-dimensional Riemannian products: see for instance the articles by Rosenberg [16], Meeks and Rosenberg [9] and Manzano, Plehnert and Torralbo [8]. In higher-dimensional manifolds constructions are also abundant.

Much work has been done considering minimal hypersurfaces which are invariant by large groups of isometries. Very closely related to this work is the article by Alencar [1] where the author studies minimal hypersurfaces in ℝ2m invariant by the action of SO(m) × SO(m). The general case of SO(m) × SO(n)-invariant minimal hypesurfaces in ℝm+n was later treated by Alencar, Barros, Palmas, Reyes and Santos in [2]. They give a complete description of such minimal hypersurfaces when m, n ≥ 3. Of great interest for the present work is also the article by Pedrosa and Ritoré [14] where the authors study the isoperimetric problem in the Riemannian products of n-dimensional simply connected spaces of constant curvature and circles. The isoperimetric regions in these spaces are known to exist (see Almgren [3] and Morgan [10; 11]) and their boundaries are hypersurfaces of constant mean curvature which are invariant by the action of the orthogonal group acting on the n-dimensional factor with constant curvature (fixing a point). Therefore in the study of the isoperimetric problem in such regions one is naturally led to study invariant hypersurfaces of constant mean curvature. A detailed study of such hypersurfaces, in particular minimal ones, was carried out by Pedrosa and Ritoré in their article.

When studying minimal hypersurfaces invariant by the group actions as in the articles mentioned above one is essentially dealing with the solutions of an ordinary differential equation. The same is true for hypersurfaces in ℝn × Sm that are invariant by the canonical action of O(n) × O(m), which is the situation we study in the present article. The main technical difference when replacing the circle by higher-dimensional spheres is that in the first case due to the invariance of the problem by rotations of the circle the associated ordinary differential equation has a first integral, which helps to describe its solutions: we have not found a first integral in the case n, m ≥ 2.

The first and main goal of this article is to classify minimal hypersurfaces in ℝn × Sm invariant by the canonical action of O(n) × O(m). There is one canonical such minimal hypersurface given by the product of ℝn with a maximal hypersphere Sm − 1Sm. The orbit space of the action of O(n) × O(m) in ℝn × Sm is identified with [0, ∞) × [0, π], and an invariant hypersurface is described by a generating curve φ in the orbit space. The mean curvature h = h(s) of the corresponding hypersurface satisfies σ(s)=(m1)cos(y(s))sin(y(s))cos(σ(s))(n1)sin(σ(s))x(s)h(s).

where φ = (x(s), y(s)) is parametrized by arc length and φ′ (s) = (cos (σ (s)), sin (σ (s))) (and h is taken with respect to the normal vector n = (sin(σ (s)), −cos (σ (s))). We will describe the invariant minimal (or constant mean curvature) hypersurfaces by studying solutions to this equation with h = 0 (or a nonzero constant).

We will give a complete description of invariant minimal hypersurfaces:

Theorem 1.1

Let n, m ≥ 2 and consider hypersurfaces inn × Sm invariant by the action of O(n) × O(m). There is a one dimensional family of invariant minimal embeddings ofn × Sm − 1 parametrized by r ∈ (0, π). There is a 2-dimensional family of invariant minimal immersions (with self-intersections) of Sn − 1 × Sm − 1 × ℝ parametrized by (r, s) ∈ (0, ∞) × (0, π). There are two 1-dimensional families of invariant minimal embeddings of Sn − 1 × ℝm parametrized by r ∈ ℝ> 0 (each family is the reflection of the other aroundn × Sm − 1). These are all minimal hypersurfaces inn × Sm invariant by the action.

Reflection around ℝn × Sm − 1 ⊂ ℝn × Sm sends the minimal immersion corresponding to the point (r, s) ∈ (0, ∞) × (0, π) to the one corresponding to (r, πs). It is easy to check that beyond these pairs the immersions corresponding to points (r1, s1) ≠ (r2, s2) are not congruent. Note also that the embeddings corresponding to r ∈ (0, π) and πr are congruent by the same reflection.

We use similar techniques to show the existence of families of embedded constant mean curvature hypersurfaces invariant by the action of O(n) × O(m). In this case the situation is considerably more complicated and we will not give a complete description. One of the main reasons why one is interested in understanding invariant hypersurfaces of constant mean curvature is to compute the isoperimetric function of ℝn × Sm. By using the Ros Product Theorem (see [15, Proposition 3.6] or [12, Section 3]) one sees that for an isoperimetric region U ⊂ ℝn × Sm the slices U ∩ (ℝn × {x}) and U ∩ ({y} × Sm) are geodesic balls centered at some fixed points p ∈ ℝn, qSm, respectively. It follows that U can be of two types: either a product of the sphere with a ball in Euclidean space or a ball type region invariant by the action of O(n) × O(m). The boundary of a region of the second type is an invariant hypersphere of constant mean curvature: the corresponding generating curve will start perpendicular to the y-axis and decrease until reaching the x-axis.

We will then concentrate on hypersurfaces starting on the y-axis, since these are the ones that could give isoperimetric regions. There is a canonical example: if xh ∈ (0, π) is defined by the equation cot(xh)=hm1 and Srm1 is the hypersphere of points at distance r from the south pole SSm, then ℝn × Sxhm1 is an invariant hypersurface of constant mean curvature h. We will prove:

Theorem 1.2

For every h ∈ ℝ> 0 there is a one-dimensional family of O(n) × O(m)-invariant embedded hypersurfaces of constant mean curvature h inn × Sm diffeomorphic ton × Sm − 1 parametrized by A ∈ (0, b) where b ∈ (xh, π]. If bπ, then there is an embedding of Sn+m − 1 with constant mean curvature h, invariant by the action of O(n) × O(m).

The isoperimetric problem in ℝn × Sm is known to have a solution for all volumes. Moreover it is known that for small values of the volume the corresponding isoperimetric region must be a ball: explicitly we point out in Lemma 5.2 that Sn − 1 (r) × Sm ⊂ ℝn × Sm is unstable (as a constant mean curvature hypersurface) if r<n1/m, where Sn − 1 (r) denotes the sphere or radius r. For instance in the case of ℝ2 × S2 this says that B(0, r) × S2 ⊂ ℝ2 × S2 cannot be an isoperimetric region for r < 1/ 2. Since the volume of B(0, 1/ 2) × S2 is 2π2 it follows from the previous comments that an isoperimetric region of volume less than 2π2 is a ball.

Therefore we know that for some (large) values of h there are invariant hyperspheres like in the second part of Theorem 1.2. Solving the equation numerically it seems that if for some value of h there exists such a hypersphere, then it is unique and it divides generating curves like the ones in Theorem 1.2 and generating curves with self-intersections. If one could prove that this is actually the case, then one would have a good understanding of the isoperimetric profile of ℝn × Sm. This should be compared to the case of spherical cylinders ℝ × Sm treated by Pedrosa in [13].

In Figure 2 we show three types of generating curves of hypersurfaces of constant mean curvature h = 1.8 that appear for the case n = m = 2. There is a value y0 ≈ 1.592 such that the generating curve of the hypersurface of constant mean curvature h starting at (0, y0) ends perpendicular to the x-axis, giving an embedded S3. The curves starting at y < y0 produce embeddings of ℝ2 × S1 and the curves starting at y > y0 produce immersions with self-intersections. In Figure 3 we still consider n = m = 2 but h = 3. Again there is a value y0 ≈ 0.98 such that the generating curve of the hypersurface of constant mean curvature h starting at (0, y0) ends perpendicular to the x-axis, giving an embedded S3. The curves starting at y < y0 produce embeddings of ℝ2 × S1 and the curves starting at y > y0 produce immersions or embeddings of constant mean curvature hypersurfaces, the latter with the property that there is a point with y > 0, x′ = 0. The corresponding hypersurface cannot be the boundary of an isoperimetric region. Note that subfigure (d) shows what would be an immersed hypersphere. Then one might conjecture non-uniqueness of constant mean curvature hyperspheres, but we have not been able to prove that this is actually the case.

Generating curves of hypersurfaces in S2 × ℝ2, with initial conditions (x0, y0, σ0).
Figure 1

Generating curves of hypersurfaces in S2 × ℝ2, with initial conditions (x0, y0, σ0).

Generating curves of minimal hypersurfaces with initial conditions (x0, y0, σ0) and constant mean curvature h = 1.8.
Figure 2

Generating curves of minimal hypersurfaces with initial conditions (x0, y0, σ0) and constant mean curvature h = 1.8.

Generating curves of constant mean curvature hypersurfaces in S2 × ℝ2, with initial conditions (x0, y0, σ0) and constant mean curvature h = 3.
Figure 3

Generating curves of constant mean curvature hypersurfaces in S2 × ℝ2, with initial conditions (x0, y0, σ0) and constant mean curvature h = 3.

Finally we will discuss the stability of the noncompact constant mean curvature hypersurfaces discussed in the previous theorems. We will prove:

Theorem 1.3

All minimal hypersurfaces inn × Sm invariant by the action of O(n) × O(m) and all the noncompact invariant constant mean curvature hypersurfaces constructed in Theorem 1.2 are unstable.

2 O(n) × O(m)-invariant hypersurfaces

We consider a hypersurface Mm+n − 1 ⊂ ℝn × Sm invariant by the canonical action of O(n) × O(m) (fixing the south pole in Sm and the origin in ℝn). The orbit space of this action is identified with [0, ∞) × [0, π], and M is identified with a curve φ in [0, ∞) × [0, π]. If we parametrize this curve by arc length, pick an orientation and write φ (s) = (x(s), y(s)), then x′(s) = cos (σ (s)) and y′ (s) = sin (σ (s)) where σ (s) is the angle formed by the (oriented) curve and the x-axis at φ (s).

The mean curvature h of M is invariant by the action of O(n) × O(m) and so it can be expressed as a function of the parameter s. It is given in terms of the curve φ and with respect to the normal vector n = (sin (σ (s)), −cos (σ (s))) by h(s)=(m1)cos(y(s))sin(y(s))cos(σ(s))(n1)sin(σ(s))x(s)σ(s).(1)

If one changes the orientation of φ and considers the curve φ (s) = φ (as) for some a ∈ ℝ, then σ (s) = σ (as) + π, and the mean curvature changes sign. Changing the orientation of φ amounts to changing the unit normal vector to M. The curve φ determines a smooth complete embedded hypersurface if and only if it does not intersect itself, it is closed and it is orthogonal to the boundary of [0, ∞) × [0, π] at points of intersection.

Except at the points where x′(s) = 0 we can also express M by a function p defined in some subset of [0, ∞) with values in [0, π] by the relation φ (s) = (t, p(t)). In terms of p, M has mean curvature h if p(x)=(1+p(x)2)((m1)cos(p(x))sin(p(x))(n1)p(x)xh1+p(x)2).(2)

Similarly at points where y′(s) ≠ 0 one can express M by a function f defined on some subset of (0, π) with values in [0, ∞) by the relation φ (s) = (f(t), t). Of course f is the inverse function of p. In terms of the function p, M has mean curvature h if f(x)=(1+f(x)2)((m1)cos(x)sin(x)f(x)+(n1)1f(x)h1+f(x)2).(3)

Note that in (2) we are parametrizing φ so that x′(s) > 0 while in (3) it is parametrized so that y′(s) > 0.

We assume from now on that h is a constant and we try to find invariant hypersurfaces of mean curvature h by studying the solutions of Equation (2).

Let xh ∈ (0, π) be the only value such that h = (m − 1) cot(xh). We have the constant solution p = xh. The first elementary observation about Equation (2) is that if at some x one has p′(x) = 0, then x is a local minimum of p if p(x) < xh, x is a local maximum of p if p(x) > xh and p is the constant solution if p(x) = xh.

We need the following completely elementary observation:

Lemma 2.1

Let j : [a, b] → [m, M] and k : [c, d] → [m, M] be two increasing C1-functions with the same image, let I = k−1j : [a, b] → [c, d] and let x0 ∈ (a, b) be such that j′(x0) = k′ (I(x0)).

  1. If j′(x0) > 0, then we have:

    1. If j″(x0) > k″ (I(x0)), then there exists ε > 0 such that j′(x) < k′(I(x)) for x ∈ (x0ε, x0) and j′(x) > k′(I(x)) for x ∈ (x0, x0 +ε).

  2. If j′(x0) = 0 but |xx0 | > 0, j′(x) > 0, k′(I(x)) > 0, then we have:

    1. If j″(x0) = k″(I(x0)) < 0 and j″′(x0) < k″′(I(x0)), then there exists ε > 0 such that j′(x) < k′(I(x)) for x ∈ (x0ε, x0).

    2. If j″(x0) = k″(I(x0)) > 0 and j″′(x0) > k″′(I(x0)), then there exists ε > 0 such that j′(x) > k′(I(x)) for x ∈ (x0, x0 + ε).

Proof

We have kI = j and so k(I(x))I(x)=j(x)k(I(x))I(x)2+k(I(x))I(x)=j(x)k(I(x))I(x)3+3k(I(x))I(x)I(x)+k(I(x))I(x)=j(x)

When j′(x0) > 0 it follows from the first equation that I′(x0) = 1. In Case 1 (a) we have that I″(x0) > 0. It follows that there exists ε > 0 such that I′(x) < 1 for x ∈ (x0ε, x0) and I′(x) > 1 for x ∈ (x0, x0 +ε) and Case 1 (a) follows.

In Case 2 (a) or 2 (b) it follows from the second equation that I′ (x0) = 1, and then it follows from the third equation that I″ (x0) > 0. Then 2 (a) and 2 (b) follow as in the Case 1 (a). □

The following two lemmas are the main tool to prove Theorem 1.1 and Theorem 1.2 in the next sections.

Lemma 2.2

Let p be a solution of (2) such that there are points x0 < x1 where x0 is a local maximum of p and x1 a local minimum, p is strictly decreasing in [x0, x1], p(x0) = E, p(x1) = e. Then p has a local maximum at some point x2 > x1 (p strictly increasing in (x1, x2)) and p(x2) < E.

Proof

Let f(x) = p(2 x1x). Then f′(x) = −p ′ (2x1x) and f″(x) = p ″ (2x1x). It follows that f satisfies f(x)=(1+f(x)2)((m1)cos(f(x))sin(f(x))+(n1)f(x)2x1xh1+f(x)2)(4)

with f(x1) = e and f′ (x1) = 0. Also f″ (x1) = p ″ (x1) = (m − 1) cos (e)/ sin (e) −h > 0. But it is easy to check that f″′ (x1) − p″′ (x1) = 2(n − 1) f″(x1) /x1 > 0. Since both f and p are strictly increasing after x1 with f′ > p′ at least close to x1 we have that for any x > x1, x close to x1, there exists a value xp close to x1, x1 < x < xp, such that f(x) = p(xp). For these values one has that f′(x) > p′(xp) by Lemma 2.1 (c).

We know that f is increasing in the interval [x1, 2x1x0] and f(2x1x0) = E. Suppose p also increases after x1 until it reaches the value E at some point xE. Then for each x ∈ (x1, 2x1x0) there exists a unique xp ∈ (x1, xE) such that f(x) = p (xp). We have seen that f′(x) > p′ (xp) for x close to x1. Suppose that there exists a first value x < 2x1x0 such that f′(x) = p′ (xp). Looking at Equations (2) and (4) one has f″(x) > p ″ (xp). But this would imply by Lemma 2.1, Case 1 (a), that for y < x, y close to x one would have f′(y) < p′ (yp), contradicting the assumption that x was the first value where the equality holds. It follows that there exists a first value z > 2x1x0 such that p′(z) = 0, p(z) = E. Then p″(z) = f″(2x1x0) = (m − 1) cos (E)/ sin (E) − h < 0 and f″′(2x1x0) − p″′(z) = p″(z) (1/(2x1x0) +1/z) < 0. Then by Lemma 2.1 (2.a) we would have f′(x) < p′(xp) for some x < 2x1x0, which we saw cannot happen. It follows that p must reach a local maximum before reaching the value E, as claimed in the lemma. □

Similarly, one has:

Lemma 2.3

Let p be a solution of (2) such that there are points x0 > x1 where x0 is a local maximum of p and x1 a local minimum, p(x0) = E, p(x1) = e. Then p has a local minimum at a point x2 > x0 (p strictly decreasing in (x0, x2)) and p(x2) > e.

The proof of Lemma 2.3 is almost the same as the one of Lemma 2.2, and we will not include it.

3 Minimal hypersurfaces

In this section we prove Theorem 1.1. The following elementary observation is needed later:

Lemma 3.1

Let f : [a, b) → [c, d] be a C2 function with a≥ 0 and f′(x) > 0 for all x ∈ (a, b), and let h:(c, d) → ℝ be a C1 function such that h′ < 0 and limxd h(x)(dx) ≤ −1. Assume that f(x)(1+f(x)2)h(f(x)).(5)

Then limxb f(x) < d.

Proof

Assume that limxb f(x) = d. Let x0 ∈ (a, b) be such that h(f(x)) (df(x)) < −1/2 for all x ∈ [x0, d). Then we have that h(f(x)) < 12(df(x)) for x ∈ [x0, d). Let ε = df(x0) > 0. Let r = f′(x0) > 0. Let δ=ε2r. For x ∈ [x0, d) we have that f″ (x) < 0 and so f′ (x) < r. Since we assume that limxb f(x) = d we must have that r(bx0) > ε. Then δ < (1/2) (bx0). Moreover, f(x0 + δ) < f(x0) + = dε /2. Also for x > x0 we have f(x)<12(df(x))<12ε, and so f(x0+δ)<r14r. The step in which one goes less than half the distance between x0 and b can be repeated any number of times. But after doing it a finite number of times one would get that f′ becomes negative, contradicting the hypothesis. Therefore limxb f(x) < d as claimed. □

Proof of Theorem 1.1

Assume that the curve φ determines a complete immersed connected minimal hypersurface. We write φ (s) = (x(s), y(s)) and denote by σ (s) the angle function as in the previous section. Then σ(s)=(m1)cos(y(s))sin(y(s))cos(σ(s))(n1)sin(σ(s))x(s).(6)

The first observation is that if φ (s) = (x(s), y(s)) determines a minimal hypersurface, then so does φ (s) = (x(s), πy(s)). Let xI = inf x(s). There are three distinct possibilities:

  • P1.

    xI = 0.

  • P2.

    xI > 0 and there is a point (xI, y) in φ with y ≠ 0, π.

  • P3.

    xI > 0 and there is a point (xI, y) in φ with y = 0 or y = π.

A priori P2 and P3 might not exclude each other but we will see that in fact they do.

Consider first the case P1. So we assume that the curve φ starts at the y-axis, i.e. it contains a point (0, A) with A ∈ [0, π]. By the previous comments we can assume that A ∈ [0, π/2]. If A = π/2, then we have the constant solution φ (s) = (s, π/2), which corresponds to σ = 0 and M = ℝn × Sm − 1.

When A = 0 the corresponding hypersurface is not smooth, it has a singularity at the point V which corresponds to (0, 0) in the orbit space (a punctured neighborhood of that point would be diffeomorphic to Sm − 1 × Sn − 1 × ℝ). One can probably study such a singular minimal hypersurface as in [1, Theorem 4.1], but we will not do it here.

Therefore we can assume that A ∈ (0, π/2). We then have a curve φ (s) = (x(s), y(s)) with φ (0) = (0, A), y′(0) = 0, x′(0) = 1. It follows that M can be described (close to this point at least) by a function p satisfying p(x)=(1+p(x)2)((m1)cos(p(x))sin(p(x))(n1)p(x)x),(7)

with initial conditions p(0) = A, p′(0) = 0 (and therefore p(0)=m12cos(A)sin(A)>0).

The proof of Theorem 1.1 is based on the following:

Proposition 3.2

Let p be a solution of (7), let z ≥ 0 and p(z) = A ∈ (0, π). Then p is defined for all t ∈ [z, ∞), p(t) ∈ (0, π), and p oscillates around π/2.

Proof

Note that if p′(x) = 0, then p has a local maximum at x if p(x) > π/2 and p has a local minimum at x if p(x) < π/2. We can assume that p′(z) ≥ 0 and p′(x) > 0 for x > z close to z (if not, we consider p = πp). We want to show that there exists x1 > z such that p′(x1) = 0.

Let [z, xF) be the maximal interval of definition of p and assume that p′ > 0 in this interval. Let yF = limxxF p(x). Lemma 3.1 tells us precisely that it cannot happen that xF < ∞ and yF = π. Also if yF > π/2, then there is a final interval where p″ has a negative upper bound. It would then follow that xF < ∞, and p is increasing and p′ decreasing close to xF; so both have limits and p could be extended beyond xF. Then we must have yFπ/2. In the same way, if xF < ∞ and if there is a final interval (x, xF) where p″ does not change sign, then p could be extended beyond xF. But if p″ keeps changing signs when x approaches xF < ∞, then the lengths of the intervals where p″ has a fixed sign would approach 0 (as we approach xF). It is easy to see that p′ (x) must be bounded and then also p″ must be bounded; it should then be clear again that limxxF p′(x) would exist and p could be extended beyond xF. We are left to assume that xF = ∞ and yFπ/2. It is clear that there must be points converging to ∞ where p″ ≤ 0. Assume that there is a point x0>m1n1 such that p″(x0) = 0. Then it follows from Equation (7) that p(x0)=(1+p(x0)2)p(x0)((m1)sin2(p(x0))+n1x02)<0.

It follows that p″(x) < 0 for all x > x0. Therefore there must exist x0 > 0 such that p″(x) < 0 for all x > x0. This implies that limx → ∞ p′(x) = 0 and then looking at Equation (7) one sees that yF = π/2. Then we can find ε > 0 very small and x > 100 (n − 1), x > x0, such that 1 + p′(x)2 < 2 and cos (p(x))/ sin (p(x)) = ε. Then p′(x) ≥ 100 ε (from Equation (7), since p″(x) < 0). For y ∈ (x, x+1) we have p″(y) ≥ −(1/50) p′(x), and then p′(x+1) > p′(x) −(1/50) p′(x) > 50 ε. It follows that p(x+1) > p(x) + 50 ε > π/2; note that for ε small enough if cos (p(x))/ sin (p(x)) = ε, then p(x) > π/2 − 2ε. This is again a contradiction and it follows that there exists a first value x1 > z which is a local maximum of p.

The same argument can now be used to show that there must be a first value x2 > x1 which is a local minimum of p. Then p will oscillate around π/2. But moreover it follows from Lemma 2.2 and Lemma 2.3 that the local maxima and minima stay bounded away from π and 0 (respectively). If the values of p at the local extrema also stay bounded away from π/2, it is elementary and easy to see from Equation (7) that the distance between consecutive extrema of p will have a positive lower bound and therefore p would be defined for all x > z. The only possibility left would be that there exists x0 > 0 such that limxx0 p(x) = π/2. But then again one would have that limxx0} p′(x) = 0 and it would follow that p must be the constant solution. This finishes the proof of the proposition. □

Coming back to the case P1 we choose a small z > 0 and apply Proposition 3.2 to see that the solution p of Equation (7) determines a complete embedded minimal hypersurface (diffeomorphic to ℝn × Sm − 1).

In the case P3 we can consider for instance the case when there is a point (xI, 0) in φ. Then we choose z > xI close to xI and again apply Proposition 3.2 to see that the corresponding solution p of Equation (7) determines a complete embedded minimal hypersurface (diffeomorphic to Sn − 1 × ℝm).

Finally in case P2 we can assume that we have a point (xI, y0) in φ with y0 ∈ (0, π/2). Then we have two branches of φ coming from the point, and each one can be described by a function p. One of them will be increasing and the other decreasing after xI. For each of the branches we can apply Proposition 3.2 to see that the corresponding solution p of Equation (7) is defined on (xI, ∞) and oscillates around π/2. It follows that one has a minimal immersion of Sm − 1 × Sn − 1 × ℝ with self-intersections.

This completes the proof of Theorem 1.1.

4 O(n) × O(m)-invariant constant mean curvature hypersurfaces

In this section we prove the existence of some invariant constant mean curvature hypersurfaces by studying the solutions of Equation (2), with h a positive constant. The equation is considerably more complicated than Equation (7): we will not be able to give such a clear description in this case. The first observation is that the isoperimetric problem on ℝn × Sm is known to have a solution for all volumes and it is easy to see by standard symmetrization arguments that the hypersurfaces which are the boundaries of the isoperimetric regions are O(n) × O(m)-invariant. And as usual they have constant mean curvature. For small values of the volume the corresponding isoperimetric region will be a ball bounded by a constant mean curvature hypersurface which will be an O(n) × O(m)-invariant sphere. This will be given by a solution of Equation (2), for some value of h, with p(0) = A > 0 and p(x) = 0 at some value x > 0. So we know that the situation will in general be different to what happened for the case of minimal hypersurfaces studied in the previous section.

As in Section 2 we let xh ∈ (0, π/2) be the value such that (m − 1)cot(xh) = h. The constant function p = xh is clearly a solution. Consider solutions p of Equation (2) with initial conditions p(0) = A, p′(0) = 0. We write p = p(A, x). Let w(x)=p(A,x)A(xh,x).

Then w satisfies w(0) = 1, w′(0) = 0 and w(x)=m1sin2(xh)wn1xw.(8)

We need the following elementary lemma:

Lemma 4.1

Let w solve Equation (8) with the initial conditions w(0) = 1, w′(0) = 0. Then there exists a sequence 0 < x1 < x2 < … such that for all i = 1, 2, …, x2i − 1 is a local minimum of (8) and x2i is a local maximum.

Proof

Let A=m1sin2(xh)) and B = n − 1. We will only use the facts that A and B are positive constants. The initial conditions imply that w″(0) < 0. Assume that w has no local minimum. Then we must have w′(x) < 0 for all x > 0. If for some y > 0 we have w(y) < 0, then we have w″(x) > A(−w(y)) (for x > y), and w′ must vanish after y. Therefore we can assume that w is always decreasing and positive. Note that w(x)Bx2w(x)+Bxw(x)+Aw(x)=0.

If for some y>B/A we have w″ (y) = 0, then it follows that w″′(y) > 0. It follows that there exists z0 > 0 such that w″ does not change sign after z0. If w″(x) < 0 for all x > z0, then there would exist y such that w(y) < 0 and we would reach a contradiction as before. If w″ is positive after z0, then it follows from (8) that for all x > z0 w(x)>ABz0w(x),

and it would follow again that there must exist y > 0 such that w(y) < 0.

Let then x1 be the first local minimum of w. Repeating the same argument we would prove that there exists a local maximum x2 > x1. And in the same way one obtains the whole sequence xi, i = 1, 2, 3, … □

Proof of Theorem 1.2

It follows from Lemma 4.1 that for A close enough to xh the corresponding solution p(A, x) of Equation (2) must have a local minimum at a value x1 > 0. Now we apply Lemma 2.2 to show that there exists x2 > x1 such that p is increasing in (x1, x2) and x2 is a local maximum of p. Then by applying Lemma 2.3 and Lemma 2.2 we see that there exists a sequence of consecutive local maxima and minima x1 < x2 < x3 < x4 < … such the sequence of local maxima p(x2i) is decreasing (and bounded below by xh) and the sequence of local minima p(x2i +1) is increasing (and bounded above by xh). Assume that one of the limits of these monotone sequences is not xh, for instance lim p(x2i) = y > xh. If the maximal interval of definition of p(A, x) were a finite interval (0, xf), then consider the solution t of the equation t(x)=(1+t(x)2)((m1)cos(t(x))sin(t(x))(n1)t(x)xfh1+t(x)2)

with initial conditions t(0) = y, t′(0) = 0. Let r > 0 be the first value such that t(r) = xh. Then for each s < r for all i big enough x2i +1x2i > s. This is a contradiction and therefore p would be defined on (0, ∞). If the limit of both monotone sequences is xh and the maximal interval of definition of p is a finite interval (0, xf), then we would have that limxxf p(x) = xh and limxxf p′(x) = 0. It would follow that p must be the constant solution and we would reach a contradiction again. It follows that p is defined on all of ℝ> 0 and gives an embedded hypersurface of constant mean curvature h.

We have proved that there is an open interval containing xh such that for all A in the interval p(A, x) determines an embedding of ℝn × Sm − 1 of constant mean curvature h. Now assume that xh > A > 0 and the corresponding solution p(A, x) does not determine such an embedding. Then from the previous discussion it follows that p(A, x) is an increasing function in a maximal interval of Definition (0, xF). Let yF = limxxF p(A, x). It follows from Lemma 3.1 that xF < ∞ and yF = π cannot happen. If yF > xh, then p″(A, x) < 0 for x close to xF. It follows that limxxF p′(A, x) exists and xF < ∞. Then p(A, x) could be extended beyond xF, reaching a contradiction. Hence we can assume that yFxh. If xF < ∞ and there is a sequence of points xi approaching xF where p″ (xi) = 0, one can see from Equation (2) that p′ must stay bounded. Then p″ must also stay bounded and limxxF p′(A, x) exists. This would imply again that p(A, x) could be extended beyond xF. The same conclusion can be reached if p″ has a constant sign close to xF. It follows that xF = ∞ and therefore p(A, x) determines an embedding of ℝn × Sm − 1 of constant mean curvature h.

Finally let b < π and assume that for all a with xh < a < b the corresponding solution p(a, x) determines an embedding of ℝn × Sm − 1 of constant mean curvature h but this is not true for p(b, x). Then from the previous discussion it follows that p(b, x) is a decreasing function in a maximal interval of Definition (0, xF).

Let yF = limxxF p(b, x).

If yF > 0, then it is elementary to see that if liminf p′(b, x) > −∞, then p″(b, x) is bounded and so the limit limxxF p′(x) exists and is finite. It then follows that the solution p(b, x) can be extended beyond xF. Then we must have that limxxF p′(x) = −∞. This corresponds to the situation when x′(s) = 0 in Equation (1). One can also study the solution by considering Equation (3) instead. The inverse function f = p−1 satisfies f′(yF) = 0 and can be extended to an interval containing yF. It then follows that for values close to b the corresponding solution of Equation (1) has the same behavior. This contradicts the fact that for every a < b the solution p(a, x) decreases until it reaches a local minimum.

It follows that yF = 0. Consider the function q(x)=p(x)1+(p(x))2=(m1)cos(p(x))sin(p(x))(n1)p(x)xh1+(p(x)2.

At a point x0 at which p″(x0) = 0 we have that q(x0)=p(x0)(m1sin2(p(x0))+n1x02).

Then for x0 close to xF we would have q′(x0) > 0. It follows that q and p″ are negative before x0 and positive after x0. Therefore p″ must have a constant sign close to xF. Hence limxxF p′(x) = L exists. If L is finite, then it is clear from Equation (2) that p″ must be positive close to xF. Then there exists x1 close to xF such that for x ∈ (x1, xF) we have p″(x) > 1/(2p(x)). Since L is finite the speed at which p reaches 0 is bounded. The previous inequality would imply that p′ must approach −∞. Hence limxxF p′(x) = −∞ and so the solution p(b, x) determines an embedding of Sn+m − 1 of constant mean curvature h.

5 Stability of the O(n) × O(m)-invariant constant mean curvature hypersurfaces

We now consider the stability of the constant mean curvature hypersurfaces described in the previous sections. We prove the instability of the O(n) × O(m)-invariant noncompact constant mean curvature hypersurfaces considered in Theorem 1.1 and Theorem 1.2. The arguments for instability go along the lines of the ones given by Pedrosa and Ritoré in [14], see also [18].

One says that an immersion j:Σk − 1Mk of a hypersurface with constant mean curvature is stable if and only if QΣ(u) ≥ 0 for all differentiable functions u:Σk − 1 → ℝ with compact support such that ∫Σ u dA = 0; see [4]. Here the index form QΣ(u) is given by QΣ(u)=Σ{u2(Ric(N)+|B|2)u2}dΣ,(9)

where N is a unit vector normal to Σ, is the volume element on Σ, Ric(N) is the Ricci curvature of N, and |B| is the norm of the second fundamental form B of Σ. This is also written as QΣ(u)=ΣuLudΣ,(10)

where L is the Jacobi operator L(u) = Δ u +(Ric (N) + |B|2) u.

Let Σ ⊂ ℝn × Sm be a hypersurface invariant by the O(n) × O(m) action and generated by a curve φ (t) as in Section 2 which satisfies x(t)=cos(σ(t))y(t)=sin(σ(t))σ(t)=(m1)cot(y(t))cos(σ(t))(n1)sin(σ(t))x(t)h.(11)

If h is constant, then direct computation gives the following: Ric(N)=(m1)cos2(σ),|B|2=σ2+(m1)cot2(y)cos2(σ)+(n1)x2sin2(σ),dΣ=xn1sinm1(y)dωm1dωn1dt,(12)

where d ωm is the volume element of the m-sphere and Δu=u+((n1)x(t)x(t)+(m1)cot(y(t))y(t))u

for an invariant function u(t). Hence we can rewrite the index form for hypersurfaces generated by solutions of (11) on an invariant function u(t), t ∈ (t0, t1), as QΣ(u)=ωm1ωn1t0t1uLuxn1sinm1(y)dt,

with Lu=u+((n1)xx(t)+(m1)cot(y)y(t))u=+((m1)cos2(σ)+σ2+(m1)cot2(y)cos2(σ)+(n1)x2sin2(σ))u.(13)

There are two canonical examples of invariant hypersurfaces of constant mean curvature h in ℝn × Sm: the product Σh1 = Sn − 1 × Sm of a sphere of constant mean curvature h in ℝn with Sm and the product Σh2 = ℝn × Sm − 1 of ℝn and a hypersphere of mean curvature h in Sm. In terms of Equation (11) they are given by the constant solutions φ (t) = ((n − 1)/h, t) and φ(t) = (t, xh), respectively. We first consider these two cases:

Lemma 5.1

For m, n > 1, the hypersurface Σh2 given by the constant solution of Equation (11) (with h constant), φ(t) = (t, xh), is unstable.

Proof

Let m, n > 1. An invariant function u: Σh2 → ℝ is a radial function on ℝn. We have QΣh2(u)=ωm1sinm1(xh)Rnu2ku2dx

with k=((hm1)2+1)(m1). Choose any u ≠ 0 with compact support such that ∫n u = 0. Then for each α > 0 the functions uα (x) = u(α x) all have mean 0 and one can pick α so that QΣh2(uα)<0; this is just the fact that the bottom of the spectrum of ℝn is 0. □

Lemma 5.2

For m, n > 1, the hypersurface Σh1 given by the constant solution of Equation (11) (with h constant), φ(t) = ((n − 1)/h, t), is unstable if and only if h>m(n1).

Proof

Note that Σh1 = Sn − 1(r) × Sm, where Sn − 1(r) denotes the sphere of radius r and r2 = (n − 1)2/h2. Thus for a function u: Σh1 → ℝ with mean 0, we get QΣh1(u)=Σh1(u2h2n1u2)dΣh1.

Hence the instability condition is equivalent to Σh1||u||2dΣh1/Σh1u2dΣh1<h2n1 for some u with mean 0. We recall that the first eigenvalue λ1 of the positive Laplacian on Σh1 satisfies λ1=inf{u:u=0}{Σh1||u||2dΣh1Σh1u2dΣh1}.

Thus the instability condition is equivalent to λ1<h2n1.

For Σh1 = Sn − 1(r) × Sm we have λ1=min{m,n1r2}=min{m,h2n1}. We conclude that Σh1 is unstable if and only if m<h2n1.

We are now ready to prove Theorem 1.3.

Proof of Theorem 1.3

Consider a hypersurface Σ that belongs to one of the families of noncompact constant mean curvature hypersurfaces described in Theorem 1.1 or Theorem 1.2. Let f(s) = (x(s), y (s), σ (s)) be the solution of Equation (11) that generates Σ. We have seen in the previous sections that in all the cases considered Σ is described by a curve which has at least one end which is given by the graph of a function p satisfying Equation (2). We have seen that p has a sequence of maxima and minima as x → ∞.

Let {(x1, y1), (x2, y2), (x3, y3), …} be the set of alternating maxima and minima of f(s), p(xi) = yi. Consider the function u = sin(σ). Direct computation yields uLu=(n1)y2x2,

where L is the operator given by Equation (13). Of course, u can be extended by symmetry to a field on all of Σ. It follows that QΣ(u)=ΣuLudΣ<0.

We next note that u vanishes at the set of alternating maxima and minima, and consider u1(x)=u(x)if x[x1,x2]0otherwise(14)

and similarly, u2(x)=u(x)if x[x2,x3]0otherwise(15)

These two functions have disjoint supports and satisfy QΣ(ui) < 0, i = 1, 2. It follows that by taking a linear combination of the two, u = C1u1+C2u2, we can construct a function such that ∫Σu = 0 and QΣ(u) < 0.

Acknowledgements

The authors would like to thank the anonymous referee for a careful reading of the original version of the manuscript and many helpful comments which were used to improve it.

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About the article

Received: 2015-02-21

Revised: 2016-06-09

Published Online: 2018-03-20

Published in Print: 2019-01-28


Communicated by: P. Eberlein

Funding: The authors are supported by grant 220074 of Fondo Sectorial de Investigación para la Educación SEP-CONACYT.


Citation Information: Advances in Geometry, Volume 19, Issue 1, Pages 1–13, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0060.

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