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Bernstein-type theorems in hypersurfaces with constant mean curvature

Abstract

By using the nodal domains of some natural function arising in the study of hypersurfaces with constant mean curvature we obtain some Bernstein-type theorems.

Riemannian manifold; eigenvalue; hypersurface; mean curvature


Bernstein-type Theorems in Hypersurfaces with Constant Mean Curvature

MANFREDO P. DO CARMO1 and DETANG ZHOU2,3

1 IMPA, Estrada Dona Castorina, 110 - Jardim Botanico, 22460-320 - Rio de Janeiro, Brazil

2 Department of Mathematics, Shandong University, Jinan, Shandong 250100, China

3 Universidade Federal Fluminense, Instituto de Matemática - 24020-140 Niterói, Brazil

Manuscript received on February 21, 2000; accepted for publication on May 9, 2000;

contributed by MANFREDO DO CARMO* * Member of the Academia Brasileira de Ciências Correspondence to: Manfredo do Carmo E-mail: manfredo@impa.br

ABSTRACT

By using the nodal domains of some natural function arising in the study of hypersurfaces with constant mean curvature we obtain some Bernstein-type theorems.

Key words: Riemannian manifold, eigenvalue, hypersurface, mean curvature.

0. INTRODUCTION

The Bernstein theorem on minimal surfaces in the Euclidean space states that if is a graph over a plane of which is defined for all points of then is itself a plane. This beautiful result has been the basis of a large number of investigations on minimal surfaces. Among its generalizations is a theorem proved independently by (do Carmo & Peng 1979) and (Fischer-Colbrie & Schoen 1980) which states that if is complete and stable then it is a plane.

A generalization of this theorem for higher dimensions was obtained by (do Carmo-Peng 1980) as follows:

THEOREM A. Let be a minimal hypersurface. Assume that is stable, complete and that

Then is a hyperplane in .

Here is the second fundamental form and is a geodesic ball of radius ball centered at some fixed point in .

Theorem A has been recently extended to hypersurfaces with constant mean curvature. A crucial point is to replace by the traceless second fundamental form ; here is the mean curvature of . The precise statement is as follows:

THEOREM B. (Alencar & do Carmo 1994a). Let

be a complete noncompact hypersurface with constant mean curvature . Assume that is strongly stable (see definition in Section 1), and that

(0.1)

Then is a hyperplane in .

In the present paper, we extend Theorem B in two directions. First we relax the growth condition on and extend Theorem B to this weaker condition. More precisely, we prove

THEOREM 1. Let be a strongly stable complete noncompact hypersurface of () with constant mean curvature . If , for some positive constants , and , where depends on given in the proof, then is a hyperplane.

Next we improve the dimension condition from to and prove

THEOREM 2. Let be a strongly stable complete noncompact hypersurface of () with constant mean curvature . Assume that

Then is a hyperplane.

Theorem 1 is the main theorem of this paper and goes a long way towards getting rid of condition (0.1) in Theorem B. For its proof we need an auxiliary proposition that might be interesting by itself and states that the function on a hypersurface with constant mean curvature in has no bounded nodal domain.

1. NOTATIONS AND PRELIMINARIES

Let be a complete noncompact hypersurface in . Fix and choose a local unit normal field . Define a linear map : by

where are the tangent vector fields and is the standard connection on . The map can be diagonalized, i.e., there exists a tangent basis such that We then define the mean curvature and the square of the second fundamental form It is well known that the above objects are independent of the choices made.

If is minimal(, we say is stable if for all piecewise smooth functions with compact support, we have that

(1.1)

here is the gradient of in the induced metric.

The notion of stability has been extended to hypersurfaces with constant mean curvature as follows: is said to be strongly stable if (1.1) holds for all piecewise smooth functions with compact support. is said to be weakly stable if (1.1) holds for all piecewise smooth functions with compact support and .

Let be an isometric immersion of a complete, noncompact Riemannian -dimensional manifold into an oriented, complete, Riemannian -dimensional manifold, a smooth unit normal field along , and the value of the Ricci curvature of in the vector . Here (this is different from the normalized one). The Morse index of M is defined as follows. Let be the second order differential operator on given by

(1.2)

Associated to is the quadratic form

(1.3)

defined on the vector space of functions on that have support on a compact domain . For each such , define the index of in as the maximal dimension of a subspace where is negative definite. The index of L in is the number defined by

(1.4)

where the supremum is taken over all compact domains . It is well known that , if is weakly stable(see, for example, (Fischer-Colbrie 1985)).

In what follows we always assume that is a hypersurface in with constant mean curvature . To study the hypersufaces with constant mean curvature, it is convenient to modify slightly the second fundamental form and to introduce a new linear map by

can also be diagonalized as:

It is easily checked that , and

Thus measures how far is from being totally umbilic. For the rest of this section we follow (Alencar & do Carmo 1994a). Choosing an orthonormal principal frame , we can write

where are components of the covariant derivative of the tensor and is the sectional curvature of the plane . By Gauss formula, we conclude that

Since , it is easy to check that:

From the above, it follows that

In this case it follows by (do Carmo & Peng 1980 (2.3), (2.4)) that

By using a lemma of Okumura (see (Alencar & do Carmo 1994b) for a proof), we have

So we have finally

(1.5)

2. A RESULT ON NODAL DOMAINS

In this section we prove a result on the nodal domains of which will be needed in our proof of main theorems. We first need to recall the definition of nodal domains.

DEFINITION. An open domain is called the nodal domain of function if for and vanishes on the boundary of . We denote by the number of disjoint bounded nodal domains of .

Now we have the following lemma which follows directly from Proposition 2.2 below. We are indebted to the referee who provided its proof and corrected a mistake in our original version.

LEMMA 2.1. Let be a hypersurface in with constant mean curvature . Then

(2.1)

PROOF. Let . Then from (1.5), with , and Proposition 2.2 below the lemma follows.

PROPOSITION 2.2. Let be Riemannian manifold and be a continuous function satisfying the following inequality of Simons' type in the distribution sense

(2.2)

where is a constant and is a continuous function on .

Then has no relatively compact nodal domain.

PROOF. Suppose that admits a relatively compact nodal domain . Write and on . Thus (2.2) can be written as

Then for any Lipschitz function with support in and vanishing at , we have

Let , for some function to be determined. We obtain

For all such that , where , we set

Denote (resp. ) the set of points in with (resp. ). A simple calculation leads to

When goes to , the first term of right hand side tends to (because is integrable), while the second term is fixed. It follows that for all functions , when is close to . These functions form an infinite dimensional vector which leads to a contradiction to the fact that is relatively compact and is continuous.

3. BERNSTEIN-TYPE THEOREMS

Before proving our main theorem, we need an auxiliary proposition. Set

PROPOSITION 3.1. Let be a complete noncompact hypersurface of () with constant mean curvature () and finite index. Assume that for some positive constants , and , where is a constant that can expressed explicitly in terms of . Then .

Our Theorem 1 is a corollary of the above proposition. It is a combination of the proposition and theorems in (Alencar & do Carmo 1994a) and (do Carmo & Peng 1980). Before proving Proposition 3.1 we give the proof of Theorem 1.

PROOF OF THEOREM 1. To prove the conclusion of Theorem 1 we only need to show that by Theorem A. Otherwise , and by Proposition 3.1 we know that . This is impossible by Theorem B. Thus the proof is complete.

We now prove the proposition:

PROOF OF PROPOSITION 3.1. Introduce in the stability inequality (1.1). It has been shown in (Alencar & do Carmo 1994a) that for all ,

(3.1)

where

If has finite index then it is stable outside some ball . In (3.1), we choose ; then and

So in this case we have

It can be checked that when , we can find sufficiently small such that . So there exists a constant which can expressed in terms of such that

(3.2)

for any piecewise smooth function with compact support in . Then

(3.3)

We claim that we can choose large enough such that for all . Otherwise we can find two positive constants such that when . Thus contains a nodal domain and this contradicts Lemma 2.1.

Assume for the sake of the contradiction that . Then from our oscillation theorem in (do Carmo & Zhou 1999 Theorem 2.1) we have that for any we can find which is not identically zero and is an oscillatory solution of

Choose where is the distance function to some fixed point in . We can find and , such that and , for all . Now choose , where is a constant such that and set . It follows that

This is a contradiction which shows our conclusion.

We now give the proof of Theorem 2:

PROOF OF THEOREM 2. We can assume that ; otherwise from (do Carmo & Peng 1980) the theorem holds. Notice that in (3.1)

(3.4)

Consider the terms without in the large bracket:

(3.5)

Then, by choosing ,

(3.6)

It is easy to see that when . Thus we can always choose sufficient small such that . Notice that our choice of makes . By using Young's inequality in (3.1)

(3.7)

where is a constant (depending on and ) and can be chosen arbitrarily small. Now set and choose small enough so that and . It follows from (3.7) that

Writing , we have

where is a constant depending only on . The rest of the proof follows exactly as in (do Carmo & Peng 1980), and we find that , a contradiction.

4. SOME FURTHER RESULTS

In this section we want to give some further related results. Using the eigenvalue estimate in (do Carmo & Zhou 1999) we can get an index estimate for hypersurfaces with nonzero constant mean curvature.

Define where is the volume of geodesic ball . It is easy to see that if has polynomial volume growth.

THEOREM 4.1. If is complete noncompact hypersurface in with nonzero constant mean curvature and , then .

In order to prove this Theorem we need to use the eigenvalue estimate theorem proved by the authors in (do Carmo & Zhou 1999) which is now restated as follows.

THEOREM. Let be a complete noncompact Riemannian manifold with infinite volume and be an arbitrary compact subset of . Then

PROOF OF THEOREM 4.1. It suffices to prove that for any natural number we can find piecewise smooth functions with compact supports such that are disjoint and

Note that from (Frensel, 1996) the volume of is infinite, so from the Theorem we have:

(4.1)

for any compact set in . So we can find a compact domain such that We also have So we can find again a compact domain such that and Repeating this procedure, we can find disjoint compact domains , , , , such that .

Let be the positive first eigenfunction of on , i.e.: in and on . We now define for and for . So

(4.2)

Thus for . This shows that , for any . So

The following is an easy consequence of Theorem 4.1.

COROLLARY 4.2. If is complete noncompact hypersurface with nonzero constant mean curvature and polynomial volume growth, then . In particular, , when with the standard metric; here is a k-dimensional sphere in .

ACKNOWLEDGMENTS

This work was done while the second author was visiting Instituto de Matemática Pura e Aplicada by an associate membership scheme of TWAS-IMPA. He wishes to thank these institutions for support and IMPA for the hospitality. The authors dedicate this paper to the memory of Carlos Chagas Filho. This work was supported partially by NNSFC, CNPq and TWAS-IMPA membership.

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  • *
    Member of the Academia Brasileira de Ciências
    Correspondence to: Manfredo do Carmo
    E-mail:
  • Publication Dates

    • Publication in this collection
      05 Oct 2000
    • Date of issue
      Sept 2000

    History

    • Accepted
      09 May 2000
    • Received
      21 Feb 2000
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