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Anais da Academia Brasileira de Ciências

Print version ISSN 0001-3765On-line version ISSN 1678-2690

An. Acad. Bras. Ciênc. vol.74 no.2 Rio de Janeiro June 2002

http://dx.doi.org/10.1590/S0001-37652002000200002 

Maximum principles for hypersurfaces with vanishing curvature
functions in an arbitrary Riemannian manifold

 

FRANCISCO X. FONTENELE1 and SÉRGIO L. SILVA2

1Universidade Federal Fluminense, Instituto de Matemática, Departamento de Geometria,
rua Mario Santos Braga s/no., 24020-140 Niterói, RJ, Brazil
2Universidade Estadual do Rio de Janeiro-UERJ, Departamento de Estruturas Matemáticas-IME
20550-013 RJ, Brazil

Manuscript received on October 30, 2001; accepted for publication on December 16, 2001;
presented by MANFREDO DO CARMO

 

 

ABSTRACT

In this paper we generalize and extend to any Riemannian manifold maximum principles for Euclidean hypersurfaces with vanishing curvature functions obtained by Hounie-Leite.
Key words: maximum principle, hypersurface, rth mean curvature.

 

 

1. INTRODUCTION

In this paper we generalize and extend to any Riemannian manifold maximum principles for hypersurfaces of the Euclidean space with vanishing curvature function, obtained by Hounie-Leite (1995 and 1999). In order to state our results, we need to introduce some notations and consider some facts. Given an hypersurface Mn of a Riemannian manifold Nn + 1, denote by k1(p),..., kn(p) the principal curvatures of Mn at p with respect to a unitary vector that is normal to Mn at p. We always assume that k1(p) £ k2(p) £ ... £ kn(p). The rth mean curvature Hr (p) of Mn at p is defined by

Hr (p) = sr(k1(p),..., kn(p)), (1)

where sr  : n® is the rth elementary symmetric function. It is easy to see that sr is positive on the positive cone n = {(x1,..., xn) Î n : xi > 0,"i}. Denote by Gr the connected component of {sr > 0} that contains the vector (1,..., 1) Î n. It was proved in Gårding (1959) that Gr is an open convex cone and that

G1É G2 É ... É Gn. (2)

Moreover on Gr,  1 £ r £ n, it holds that (see Caffarelli et al. 1985, Proposition 1.1)

> 0,    1 £ i £ n. (3)

As it was observed in Hounie-Leite (1995), the subset {sr = 0} can be decomposed as the union of r continuous leaves Z1,..., Zr, being Z1 the boundary ¶ Gr of the cone Gr. Furthermore each leaf Zj may be identified with the graph of a continuous function defined in the plane x1 + ... + xn = 0. Following Hounie-Leite(1995), we say that a point x = (x1,..., xn) Î n has rank r if exactly r components of x do not vanish.

As in Fontenele-Silva (2001), given p Î Mn and a unitary vector ho that is normal to Mn at p, we can parameterize a neighborhood of Mn containing p and contained in a normal ball of Nn + 1 as

j(x) = expp(x + m(x)ho), (4)

where the vector x varies in a neighborhood W of zero in TpM and m : W ® satisfies m (0) = 0 and m(0) = 0, being the gradient operator in the Euclidean space TpM. Choosing a local orientation h : W ® M of Mn with h(0) = ho, we denote by Hr(x) the rth mean curvature of Mn at j(x) with respect to h(x).

Given hypersurfaces M and M' of Nn + 1 that are tangent at p and a unitary vector ho that is normal to M at p, we parameterize M and M' as in (4), obtaining correspondent functions m : W ® and m' : W ® , defined in a neighborhood W of zero in TpM = TpM'. As in Fontenele-Silva (2001), we say that M remains above M' in a neighborhood of p with respect to ho if m(x) ³ m'(x) for all x in a neighborhood of zero. We say that M remains on one side of M' in a neighborhood of p if either M is above M' or M' is above M in a neighborhood of p. Finally, denote by (p) = (k1(p),..., kn(p)) and by (p) = (k'1(p),..., k'n(p)) the principal curvature vectors at p of respectively M and M'.

We can now state our results:

THEOREM 1.a. Let M and M' be hypersurfaces of Nn + 1 that are tangent at p, with normal vectors pointing in the same direction. Suppose that M remains on one side of M' and that Hr(x) = H'r(x) in a neighborhood of zero in TpM, for some r,  1 £ r < n. If r ³ 2, suppose further that (p) and (p) belong to same leaf of {sr = 0} and the rank of either (p) or (p) is at least r. Then, M and M' must coincide in a neighborhood of p.

THEOREM 1.b. Let M and M' be hypersurfaces of Nn + 1 with boundaries M and M', respectively, and assume that M and M', as well as M and M', are tangent at p Î M ǶM', with normal vectors pointing in the same direction. Suppose that M remains on one side of M' and that Hr(x) = H'r(x) in a neighborhood of zero in TpM, for some r,  1 £ r < n. If r ³ 2, suppose further that (p) and (p) belong to same leaf of {sr = 0} and the rank of either (p) or (p) is at least r. Then, M and M' must coincide in a neighborhood of p.

As a consequence of Theorems 1.a and 1.b, we obtain the following corollaries, that extend Theorem 0.1 in Hounie-Leite (1995) to any Riemannian manifold.

COROLLARY 1.a. Let M and M' be hypersurfaces of Nn + 1 that are tangent at p, with normal vectors pointing in the same direction and with both having r-mean curvature equal to zero for some r, 1 £ r < n. For r ³ 2, suppose further that (p) and (p) belong to same leaf of {sr = 0} and the rank of either (p) or (p) is at least r. Under these conditions, if M remains on one side of M', then M and M' must coincide in a neighborhood of p.

COROLLARY 1.b. Let M and M' be hypersurfaces of Nn + 1 with boundaries M and M', respectively, so that M and M', as well as M and M', are tangent at p Î ¶M Ç M', with normal vectors pointing in the same direction. Assume that M and M' have r-mean curvature equal to zero for some r, 1 £ r < n. For r ³ 2, suppose further that (p) and (p) belong to same leaf of {sr = 0} and the rank of either (p) or (p) is at least r. Under these conditions, if M remains on one side of M', then M and M' must coincide in a neighborhood of p.

The extension of Theorem 1.3 in Hounie-Leite (1999) is given in the following theorems.

THEOREM 2.a. Let M and M' be hypersurfaces of Nn + 1 that are tangent at p, with normal vectors pointing in the same direction. Suppose that M remains above M' and that H'r ³ 0 ³ Hr, for some r,  2 £ r < n. Suppose further that H'j (p) ³ 0, 1 £ j£ r – 1, and either Hr+1(p) ¹ 0 or H'r+1(p) ¹ 0. Then, M and M' must coincide in a neighborhood of p.

THEOREM 2.b. Let M and M' be hypersurfaces of Nn + 1 with boundaries M and M', respectively, and assume that M and M', as well as M and M', are tangent at p Î M Ç M' with normal vectors pointing in the same direction. Suppose that M remains above M' and that H'r ³ 0 ³ Hr, for some r,  2 £ r < n. Suppose further that H'j (p) ³ 0, 1 £ j£ r – 1, and either Hr+1(p) ¹ 0 or H'r+1(p) ¹ 0. Then M and M' must coincide in a neighborhood of p.

It will be clear from the proofs that in Theorems 2.a and 2.b we only need to require H'r (x) ³ Hr (x), in a neighborhood of zero in TpM, and H'r (p) ³ 0 ³ Hr (p) instead of H'r ³ 0 ³ Hr everywhere. For r = 1, it must be observed that, in Theorems 2.a and 2.b, we can assume only that H'r (x) ³ Hr (x) and that M remains above M' in a neighborhood of zero in TpM (see Theorems 1.1 and 1.2 in Fontenele-Silva (2001)).

 

2. PRELIMINARIES

In this section we will present the necessary material for our proofs.

Following Hounie-Leite (1995), we say that x Î n is an elliptic root of sr if sr(x) = 0 and either (x) > 0,  j = 1,..., n, or (x) < 0,  j = 1,..., n. It is easy to see that any root of s1 = 0 is elliptic. For 2 £ r < n, we have the following criterion of ellipticity (see Corollary 2.3 in Hounie-Leite (1995) and Lemma 1.1 in Hounie-Leite (1999)):

LEMMA 1. Let x Î n and assume that sr(x) = 0 for some 2 £ r < n. Then, the following conditions are equivalent

(i) x is elliptic.

(ii) the rank of x is at least r.

(iii) sr+1(x) ¹ 0

For the proofs of our results, we will also need of the following lemmas:

LEMMA 2. If y, w belong to a leaf Zj of sr = 0, w - y belongs to the closure of n and either y or w is an elliptic root, then y = w.

LEMMA 3. For 1 £ r £ n, if Î n satisfies sj (x) ³ 0, 1 £ j £ r, then x Î .

Lemma 2 is a particular case of Lemma 1.3 in Hounie-Leite (1995) and Lemma 3 follows from the proof of Lemma 1.2 in Hounie-Leite (1999).

For d = (n(n + 1)/2) + 2n + 1, write an arbitrary point p at d as

p = (r11,..., r1n, r22,...r2n,..., r(n - 1)n, rnn, r1,..., rn, z, x1,..., xn)

or, in short, as p = (rij, ri, z, x) with 1 £ i £ j £ n, and x = (x1,..., xn). A C1-function F : G ®   defined in an open set G of d is said to be elliptic in p Î G if

(p)xixj > 0      for all nonzero      (x1,x2,...,xn) Î n. (5)

We say that F is elliptic in G if F is elliptic in p for all p Î G. Given a function f : U ® of class C2, defined in an open set U Ì n, and x Î U, we associate a point L(f )(x) in d setting

L(f )(x) = (fij(x), fi(x), f (x), x), (6)

where fij(x) and fi(x) stand for (x) and (x), respectively. Saying that the function F is elliptic with respect to f means that L(f )(x) belongs to G and F is elliptic in L(f )(x) for all x Î U. For elliptic functions it holds the following maximum principle(see Alexandrov 1962):

MAXIMUM PRINCIPLE. Let f, g : U ® be C2-functions defined in an open set U of n and let F : G Ì  d be a function of class C1. Suppose that F is elliptic with respect to the functions (1 - t) f + tgt Î [0, 1]. Assume also that

F(L(f )(x)) ³ F(L(g)(x))      ,"x Î U, (7)

and that f £ g on U. Then, f < g on U unless f and g coincide in a neighborhood of any point xo Î U such that f (xo) = g(xo).

Consider now a hypersurface Mn Ì Nn + 1, a point p Î M and a unitary vector ho that is normal to Mn at p. Fix an orthonormal basis e1,..., en in TpM and introduce coordinates in TpM by setting x = xiei for all x Î TpM. Parameterize a neighborhood of p in M as in (4), obtaining a function m : W Ì TpM ® . Recall that m(0) = 0 and (0) = 0, for all i, 1 £ i £ n. Choose a local orientation h : W ® M of Mn with h(0) = h0 and denote by Ah(x) the second fundamental form of Mn in the direction h(x). Denote by j(x) the vector (x) and by A(x) = (aij(x)) the matrix of Ah(x) in the basis ji(x). In Fontenele-Silva (2001), it is proved the existence of an n ´ n-matrix valued function à defined in an open set (n(n + 1)/2) + n ´ Ì d, being an open set of n + 1, containing the origin of d such that

Ã(L(m)(x)) = A(x)  ,    x Î W. (8)

Moreover, we have Ã(rij, ri, z, x) diagonalizable for all (rij, ri, z, x) Î (n(n + 1)/2) + n ´ . Consider the function Fr  :(n(n + 1)/2) + n ´ ® defined by

Fr = sroloÃ, (9)

where l(Ã(w)) = (l1(Ã(w)),...,ln(Ã(w)) for all w Î (n(n + 1)/2) + n ´ . Here l1(Ã(w)) £ ... £ ln(Ã(w)) are the eigenvalues of Ã(w). It follows from (1), (8) and (9) that

Hr(x) = Fr(L(m)(x))    ,    x Î W. (10)

The proof of Proposition 3.4 in Fontenele-Silva (2001) gives

 (rij, 0, 0, 0) xkxl =  (Ã((rij, 0, 0, 0))) xkxl, (11)

for all (rij, 0, 0, 0) Î d.

We also make use of the following lemma

LEMMA 4. If Ao Î n() is symmetric and (l(Ao)) > 0    ( < 0)     for all 1 £ i £ n, then

(Aoxixl > 0    ( < 0) ,    "  x = (x1,...,xn) ¹ 0. (12)

The proof of Lemma 4 follows from the proof of Lemma 3.3 in Fontenele-Silva (2001).

 

3. PROOFS OF OUR RESULTS

We will prove only Theorems 1.a and 2.a, since the proofs of Theorems 1.b and 2.b are analogous.

PROOF OF THEOREM 1.a. If r = 1, the theorem follows from Theorem 1.1 in Fontenele-Silva (2001). Thus, we assume that 2 £ r < n. The assumption Hr(x) = H'r(x) in a neighborhood W of zero in TpM and (10) imply that

Fr(L(m)(x)) = Fr(L(m')(x))  ,    x Î W. (13)

On the other hand, (p) and (p) are both roots of sr = 0 and one of them is elliptic by our hypothesis and Lemma 1. The fact that M remains on one side of M' implies that either (p) - (p) or (p) - (p) belongs to . Since (p) and (p) belong to same leaf of {sr = 0} by assumption, it follows from Lemma 2 that

(p) = (p). (14)

For each t Î [0, 1], if we consider the hypersurface Mt parameterized by

j(x) = expp(x + ((1 - t)m + tm')(x)ho)  ,  x Î W, (15)

we have that Mt is tangent to both M and M' in p and that Mt is between M and M' in a neighborhood of p. Using (14), we conclude that the principal curvature vector of Mt at p is equal to (p) = (p), for all t Î [0, 1]. This implies, by (8), that

l o Ã((1 - t)L(m)(0) + tL(m')(0)) = (p) = (p)  , (16)

for all t Î [0, 1]. Since (p) = (p) is elliptic, it follows from (11) and Lemma 4 that either Fr or -Fr is elliptic along the line segment (1 - t)L(m)(0) + tL(m')(0) Ì (n(n + 1)/2) + n ´ Ì d. Since ellipticity is an open condition, restricting W if necessary, we conclude by continuity and by the compactness of [0,1] that either Fr or -Fr is elliptic in (1 - t)L(m)(x) + tL(m')(x), for all t Î [0, 1] and x Î W. Consequently either Fr or -Fr is elliptic with respect to the functions (1 - t)m + tm',  t Î [0, 1]. So, by (13), we can apply the maximum principle to conclude that m and m' coincide in a neighborhood of zero. Therefore, M and M' coincide in a neighborhood of p.

PROOF OF THEOREM 2.a. By our assumptions it holds that H'r (x) ³ Hr (x) for x Î W. This and (10) imply that

Fr(L(m)(x)) - Fr(L(m)(x)) ³ 0  ,    x Î W. (17)

Since M remains above M', we have (p) - (p) Î n. It follows from our assumptions and Lemma 3 that (p) Î . We claim that (p) Î ¶Gr . Otherwise, by Lemma 4.1 in Fontenele-Silva (2001), we would have that (p) Î Gr, which is a contradiction since Hr (p) £ 0. So (p) Î Z1 = ¶Gr. We can use Lemma 4.1 in Fontenele-Silva (2001) to conclude that (p) Î Z1 = ¶Gr. As in the proof of Theorem 1.a, we can use Lemmas 1 and 2 to obtain that (p) = (p). Since > 0    on  Gr,    1 £ i £ n, (p) = (p) is an elliptic root of sr = 0 and (p) = (p) ζGr, we deduce that

((p)) > 0    ,    "i = 1,..., n. (18)

Now, proceeding as in the proof of Theorem 1.a, we conclude that is elliptic with respect to the functions (1 - t)m + tm',  t Î [0, 1]. It follows from (17) and the maximum principle that m and m' coincide in a neighborhood of zero. Therefore, M and M' coincide in a neighborhood of p

 

ACKNOWLEDGMENTS

We would like to thank M.P. do Carmo for suggesting us to publish this work.

 

 

RESUMO

Neste trabalho nós generalizamos e estendemos para uma variedade Riemanniana arbitrária princípios do máximo para hipersuperfícies com r-ésima curvatura média zero no espaço Euclidiano, obtidos por Hounie-Leite.
Palavras-chave: princípio do máximo, hipersuperfície, r-ésima curvatura média.

 

 

REFERENCES

ALEXANDROV AD. 1962. Uniqueness theorems for surfaces in the large I. Amer Math Soc Transl, Ser 2, 21: 341-354.         [ Links ]

CAFFARELLI L, NIRENBERG L AND SPRUCK J. 1985. The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the hessian. Acta Math 155: 261-301.         [ Links ]

FONTENELE F AND SILVA SL. 2001. A tangency principle and applications. Illinois J Math 45: 213-228.         [ Links ]

GåRDING L. 1959. An inequality for hyperbolic polynomials. J Math Mech 8: 957-965.         [ Links ]

HOUNIE J AND LEITE ML. 1995. The maximum principle for hypersurfaces with vanishing curvature functions. J Differ Geom 41: 247-258.         [ Links ]

HOUNIE J AND LEITE ML. 1999. Two-ended hypersurfaces with zero scalar curvature. Indiana Univ Math J 48: 817-882.         [ Links ]

 

 

Correspondence to: Francisco X. Fontenele
E-mail: fontenele@mat.uff.br

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