Minimal hypersurfaces in ,
total curvature and index

Pierre Bérard and Ricardo Sa Earp
November 2009

In this paper, we consider minimal hypersurfaces in the product space . We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.


pb1-figs \excludeversionpb2 \excludeversionpb3 \excludeversionpb4

MSC(2000): 53C42, 58C40.

Keywords: Minimal hypersurfaces, stability, index.

1 Introduction

In this paper, we focus on complete oriented minimal hypersurfaces immersed in equiped with the product metric.

In Section 3, we study the family } of hypersurfaces invariant under rotations about the vertical geodesic (“catenoids”) and the family of hypersurfaces invariant under hyperbolic translations. These examples generalize to higher dimensions some of the minimal surfaces constructed in [21, 19, 20].

In particular, we prove that the -dimensional catenoids have vertical heights bounded from above by (Proposition 3.2). In Section 3.3, we describe the maximal stable rotationally invariant domains on and we prove that the catenoids have index (Theorem 3.5). We also give an interpretation in terms of the envelope of the family (Corollary 3.7). Finally, we observe that the half-catenoid is not maximally stable.

We describe the minimal hypersurfaces invariant under hyperbolic translations in Theorem 3.8. In particular, we find a hypersurface which is a complete non-entire vertical graph over a half-space bounded by some hyperplane in . It takes infinite value data on and zero asymptotic boundary value data. When , the hypersurface is an entire vertical graph. When , it is a bi-graph over the exterior of an equidistant hypersurface of .

In Section 4, we consider the relationships between finiteness of the total curvature and finiteness of the index. In dimension , we consider the curvature integrals and , where is the second fundamental form of the immersion and the Gauss curvature. Finiteness of these integrals implies that the corresponding curvatures tend to zero uniformly at infinity; finiteness of the latter implies finiteness of the index of the Jacobi (stability) operator (Theorem 4.1). The converse statements do not hold. On the one hand, the catenoids have finite index although they have infinite total intrinsic curvature. This is in contrast with the case of minimal surfaces in Euclidean -space ([12]) and with the case of surfaces with constant mean curvature in hyperbolic -space ([11, 10]). Note that catenoids have finite total extrinsic curvature. On the other hand, the surfaces invariant under hyperbolic translations are stable graphs, their curvature goes to zero at infinity although they have infinite total curvature. The proof we give of Theorem 4.1 relies mainly on Simons’ equation and the de Giorgi-Moser-Nash method which shows that finite total curvature implies that the curvature tends to zero uniformly at infinity. We point out that the finiteness of the intrinsic total curvature has deep consequences. Under this assumption on , L. Hauswirth and H. Rosenberg ([14], Theorem 3.1) have indeed shown that the total intrinsic curvature is quantified, that the ends of are asymptotic to Scherk type surfaces and obtained a -control on the curvature at infinity. In dimension , we give an upper bound of the index in terms of the total extrinsic curvature (Theorem 4.3).

In Section 5, using the catenoids as barriers, we prove some symmetry and characterization results for minimal hypersurfaces in whose boundary consists of two congruent convex hypersurfaces in parallel slices (Theorem 5.1). We point out that the hypersurfaces ( and ) have been used in [22, 23] as barriers for the Dirichlet problem and that they play a crucial role for some existence theorem for the vertical minimal surface equation.

Finally, we point out that most of our results may be established if the ambient space is one of the product spaces or .

The authors would like to thank the Mathematics Department of PUC-Rio (PB) and the Institut Fourier – Université Joseph Fourier (RSA) for their hospitality. They gratefully acknowledge the financial support of CNPq, FAPERJ, Université Joseph Fourier and Région Rhône-Alpes.

2 General framework

2.1 Notations

We consider hypersurfaces immersed in the space equiped with the product metric , where is the hyperbolic metric,


We have chosen the ball model for the -dimensional hyperbolic space .

2.2 Jacobi operator, Index, Jacobi fields

Let be an orientable minimal hypersurface in an oriented Riemannian manifold with metric . Let be a unit normal field along and let be the second fundamental form of the immersion with respect to . Let be the normalized Ricci curvature of . The second variation of the volume functional gives rise to the Jacobi operator (or stability operator) of (see [25, 17, 8]),


where is the (non-positive) Laplacian on (for the induced metric).

Given a relatively compact regular domain on the hypersurface , we let denote the number of negative eigenvalues of for the Dirichlet problem on (this is well defined because is compact). The index of is defined to be the supremum ()


taken over all relatively compact regular domains.

Let be the least eigenvalue of the operator with Dirichlet boundary conditions in . Recall that a relatively compact regular domain is said to be stable, if ; unstable, if ; stable-unstable, if . More generally, we say that a domain is stable if any relatively compact subdomain is stable.

Properties 2.1

Recall the following properties.

  1. Let be a stable-unstable relatively compact domain. Then, any smaller domain is stable while any larger domain is unstable (monotonicity of Dirichlet eigenvalues).

  2. Of particular interest are the solutions of the equation . We call such functions Jacobi fields on . Let be a one-parameter family of oriented minimal immersions, with variation field and unit normal . Then, the function is a Jacobi field on ([1], Theorem 2.7).

  3. Let be a relatively compact domain on a minimal manifold . If there exists a positive function on such that , then is stable ([13], Theorem 1).

3 Examples of minimal hypersurfaces in

In this section we give examples of minimal hypersurfaces in . We use these examples as guidelines and counter-examples to study the relationships between index properties of the Jacobi operator and the finiteness of some total curvature of , see Theorems 4.1 and 4.3. We also use them as barriers for a symmetry and characterization result in Section 5.

3.1 Rotation hypersurfaces in

We first consider rotation hypersurfaces about a vertical geodesic axis in . Up to isometry, we can assume the rotation axis to be . Recall that we take the ball model for .

Take the vertical plane and consider a generating curve for some positive function which represents the hyperbolic distance to the axis , at height .

We define a rotation hypersurface by the “parametrization”


where is a point in the unit sphere and stands for the point in the ball .

The basic tangent vectors to the immersion are

where is the derivative of with respect to , and

where is a unit vector.




We collect basic formulas in the next proposition whose proof is straightforward.

Proposition 3.1

Let be an isometric immersion. We have the following formulas in the parametrization on .

  1. The induced metric is given by


    where is the canonical metric on .

  2. The Riemannian measure for the metric is given by


    where is the canonical measure on the sphere.

  3. The unit normal field to the immersion can be chosen to be


    In particular, the vertical component of the unit normal field is given by


At the point , the principal directions of curvature of are

  • the tangent to the meridian curve in the vertical -plane

  • the vectors tangent to the distance sphere at in the hyperbolic slice , where the restriction of the second fundamental form is a scalar multiple of the identity.

The principal curvatures with respect to are

  • , the principal curvature in the direction tangent to the meridian curve, given by

  • the principal curvatures in the directions tangent to at ,


We conclude that the mean curvature of the rotation hypersurface with respect to the unit normal is given by




3.2 Catenoids in

In this Section, we describe the minimal rotation hypersurfaces about , in . By analogy with the Euclidean case, we call them catenoids. They are the higher dimensional counterparts of the catenoids constructed in [21].

Given some , let denote the maximal solution of the Cauchy problem


where and are the first and second derivatives of with respect to .

Proposition 3.2

For , the maximal solution gives rise to the generating curve , (catenary), of a complete minimal rotation hypersurface (catenoid) in , with the following properties.

  1. The interval is of the form for some finite positive number and is an even function of the second variable.

  2. For all , .

  3. The derivative is positive on , negative on .

  4. The function is a bijection from onto , with inverse function given by

  5. The catenoid has finite vertical height ,

  6. The function increases from to when increases from to infinity. Furthermore, given , the generating catenaries and intersect at exactly two symmetric points.

Proof. Assertion 1 follows from the Cauchy-Lipschitz theorem for some positive which is finite as we will see below.

Assertion 2 follows from the fact that for all (see (3.12)).

Assertion 3 is clear.

Assertion 4. According to Assertion 3, is increasing so that it has a limit when tends to and this limit must be infinite because we took a maximal solution. It follows that the inverse function maps onto and that . Finally, we find that on and the formula for follows because . Note that the integral (3.14) converges at .

Assertion 5. We have that , where

where the integral converges at both and .

Assertion 6. By a change of variables, we can write

and compute the derivative

Note that

and that the right-hand side is in for , so that we can take the limits under the integral and obtain that and . The last integral can be calculated explicitly because . The last assertion follows by considering the function and by using the monotonicity of .

Remark.  The above proposition shows that the catenoids in have uniformly bounded finite vertical height. This is in contrast with the Euclidean catenoids () which have finite, yet unbounded, vertical heights. Catenaries
Figure 1: Catenaries

Notations.  Let denote the unit normal to the catenoid , let denote its second fundamental form relative to the normal and let denote its Riemannian measure. When , let denote the Gauss curvature of . We state the following proposition for later purposes.

Proposition 3.3

For , the -dimensional catenoid in has infinite volume and finite total extrinsic curvature . When , the catenoid has infinite total intrinsic curvature .

Proof. We can restrict to the upper half-catenoid, , which admits the parametrization

The geometric data of are readily calculated. In particular,


The first assertion follows ( tends to zero exponentially at infinity). For the second assertion, we use Gauss equation and minimality to get that

where is the sectional curvature of the -plane tangent to in the ambient space and where is the vertical component of the unit normal to ,

Assertion 2 follows because tends to at infinity on .

3.3 Catenoids in , stability properties

Recall that the catenoid is generated by the curve in the vertical plane , where is the maximal solution of the Cauchy problem (3.13). This yields the parametrization


for , with and . According to Property 2.1 (2), we have two Jacobi fields on the catenoid .

 The vertical Jacobi field comes from the vertical translations in . It is given by , where is the unit normal to . According to (3.7), it is given by the formula


where stands for the derivative with respect to the variable . Because is even, the function is odd.

 The variation Jacobi field comes from the variations with respect to the parameter . It is given by . According to (3.7) and (3.16), the function is given


where stands for the derivative with respect to the variable . Because is even, the function is even.

 The Jacobi fields and have nice expressions when restricted to the upper-half of the catenoid . Indeed, recall that the function has an inverse function given by (3.14). Using the relationships


we get the following expressions for and for ,


For , define the functions and , by the following formulas


From (3.14), we can write


and compute ,

We obtain,


We summarize the relevant properties in the following lemma whose proof is straightforward.

Lemma 3.4

Define the functions and for by


see Formulas(3.20). Then,


Furthermore, for ,

  1.   and   ,

  2.   and   ,

  3. for , so that
    for ,  and  .

Notation.  For , let denote the rotationally symmetric domain


In particular, is the half-vertical catenoid .

Theorem 3.5

The stability properties of the rotationally symmetric domains on the catenoid are as follows.

  1. There exists some such that the relatively compact domain is stable-unstable. Hence, for any , the domain is stable; for any , the domain is unstable.

  2. There exists some such that

    1. the (non relatively compact) domain is stable,

    2. for any , there exists some such that the domain is stable-unstable.

  3. The catenoid has index .

The above domains are generated by the portions of curves illustrated in the following figures.

Case 1
Figure 2: Case 1
Case 1
Figure 3: Case 2a
Case 1
Figure 4: Case 2b

Proof. Assertion 1. Consider the function . According to Lemma 3.4, and , so that it must vanish at least once on . It turns out (compare with Lemma 3.6 below) that has a unique positive zero . Because is even in , it does not vanish in the open set and satisfies , and This means that is a stable-unstable domain. The second assertion follows from Property 2.1 (1).

Assertion 2. Take any and define the function by


This is a Jacobi field on and furthermore , because is odd and is even with respect to . Note also that .

Lemma 3.6

The function vanishes only once on and vanishes at most once on .

Let us prove the first assertion of the Lemma, the proof of the second assertion is similar. Assume that has at least two consecutive zeroes in the interval . The domain would then be stable-unstable because on and because vanishes on . On the other-hand, the Jacobi field satisfies and in . By Property 2.1 (3), we have that which contradicts the fact that this domain is stable-unstable. This proves the lemma.

In order to determine whether the function vanishes on or not, it is sufficient to look at the behaviour of when tends to from below. For this purpose, we use the expression (3.24) for and we write

Using Lemma 3.4, we can write

If , then does not vanish on and in fact on ; if , then has one and only one zero on .

We now observe that is a Jacobi field on which take the value at and the value at . It follows from Lemma 3.6 that has one and only one positive zero . We have that on , so that for any , the function has only one zero on . This proves the Assertion 2(a). On the other-hand, on , so that for any , the function has a unique positive zero . This proves the Assertion 2(b).

Assertion 3. Assertion 1 shows that has index at least . In order to show that the index is at most one, we use Fourier decomposition with respect to the variable and an extra stability argument.

Recall that we work in the ball model for . Let be a geodesic through in . Up to a rotation, we may assume that . Let and let . We call this set a half-horizontal catenoid.

Claim 1. A half-horizontal catenoid is stable.

To prove the claim, we shall find a positive Jacobi field on .

Let denote the complex coordinate in (ball model). We consider the group of hyperbolic isometries along the geodesic and we extend these isometries slice-wise as isometries in . We then have the one-parameter group of isometries

The associated Killing vector-field in is given by or, in the coordinates, which can be written as

where and are seen as vectors in .

This formula can easily be generalized to higher dimensions as

where , and where and are seen as vectors in . Writing the point in the parametrization as , we obtain that

Using the fact that on , we find that the Killing field gives rise to the horizontal Jacobi field

which is positive on .

Claim 2. On equiped with the standard Riemannian metric, there exists an orthonormal basis of spherical harmonics with the property that the nodal domains of the are contained in hemispheres.

The property is clearly true on and can be proved by induction on the dimension, using polar coordinates centered at a given point on the sphere.

Claim 3. The Jacobi operator on can be written as

where is a Sturm-Liouville operator on the variable, with coefficients depending only on and , where is a positive function and where is the Laplacian of the sphere acting on the -variable. This claim follows immediately from Formulas (3.5) and (3.6) for the metric and the Riemannian measure on a rotation hypersurface and from the expression for the quadratic form associated with .

Assume that the index of is at least . Then, there exists some such that has at least two negative eigenvalues in (we only consider Dirichlet boundary conditions). Because the least eigenvalue is simple, a corresponding eigenfunction must be invariant under rotations (i.e. only depends on the variable ) and say positive. Consider an eigenfunction associated with . We claim that cannot be invariant under rotations. Indeed, would otherwise depend only on the variable , it would be orthogonal to and hence it would have to vanish on . This would contradict the fact that the domains and are stable. Since is not rotationally invariant, there exists some and some in the decomposition into spherical harmonics with respect to the second variable, . We would have . Using Claim 2 and the fact that , this would mean that any nodal domain of is unstable, in contradiction with Claim 1.

Assuming that the index is at least therefore yields a contradiction and hence the index of is exactly one.

Remark.   It follows from the positivity of the Jacobi field for that the upper half-catenoid is stable (in the sense that any relatively compact domain contained in is stable, see Section 2.2). The second assertion in the preceding theorem says more. Indeed, there exists some such that the non-compact domain is stable and stricly contains