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
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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
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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
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Associated to is the quadratic form
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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
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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
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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
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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
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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
,
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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
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for any piecewise smooth function with compact support in
. Then
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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)
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Consider the terms without in the large bracket:
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Then, by choosing ,
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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)
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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:
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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
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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.
- ALENCAR H & DO CARMO MP. 1994a. Hypersurfaces with constant mean curvature in space forms. An Acad Bras Ci 66: 265-274.
- ALENCAR H & DO CARMO MP. 1994b. Hypersurfaces with constant mean curvature in spheres. Proc Amer Math Soc 120: 1223-1229.
- DO CARMO MP & PENG CK. 1979. Stable complete minimal surfaces in R3 are planes. Bull Amer Math Soc N.S. 1: 903-906.
- DO CARMO MP & PENG CK. 1980. Stable complete minimal hypersurfaces. Proc of the 1980 Beijing Symp. CHERN SS & WENTSUN W (Eds.), Gordan and Breach Science Pub., 1349-1358.
- DO CARMO MP & ZHOU D. 1999. Eigenvalue Estimate on Complete Noncompact Riemannian Manifolds and Application. Trans Amer Math Soc 351: 1391-1401.
- FISHER-COLBRIE D. 1985. On complete minimal surfaces with finite Morse index in three-manifolds. Invent Math 82: 121-132.
- FISHER-COLBRIE D & SCHOEN R. 1980. The structure of complete stable minimal surfaces in 3-manifolds of nonnegative curvature. Comm Pure Appl Math 33: 199-211.
- FRENSEL KR. 1996. Stable complete surfaces with constant mean curvature. Bol Soc Bras Mat 27: 129-144.
Publication Dates
-
Publication in this collection
05 Oct 2000 -
Date of issue
Sept 2000
History
-
Accepted
09 May 2000 -
Received
21 Feb 2000