Erratum to "Bernstein-type theorems in hypersurfaces with constant mean curvature" [An Acad Bras Cienc 72(2000): 301-310]

Erratum to "Bernstein-type Theorems in Hypersurfaces with Constant Mean Curvature" [An Acad Bras Cienc 72(2000): 301-310]

MANFREDO P. DO CARMO¹ and DETANG ZHOU^{2, 3}

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

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

³Departamento de Geometria, Universidade Federal Fluminense (UFF), 24020-140 Niterói, RJ, Brazil

Manuscript received on July 24, 2001; accepted for publication on August 1, 2001.

ABSTRACT

An erratum to Lemma 2.1 in Do Carmo and Zhou (2000) is presented.

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

ERRATUM

Replace Section 2 in Do Carmo and Zhou (2000) by the following. The resulting change in the lemma will not affect the rest of the paper.

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 D is called the nodal domain of a function f if f (x) 0 for x int D and vanishes on the boundary of D. We denote by N(f ) the number of disjoint bounded nodal domains of f.

Now we have the following lemma which follows directly from Proposition 2.2 below. We want to thank the referee who provided the clearer proof of Proposition 2.2.

LEMMA 2.1. Let M be a hypersurface in R^{n +} 1 with constant mean curvature H. Then

(

(2.1)

PROOF. Let N = N(||) and D₁, D₂, ^... , D_Nbe the N nodal domains of || and let

j(u) = u² + Hu - nH².

Then from (1.5) and Proposition 2.2 below we have functions f₁, f₂, ^... , f_N with supports in D₁, D₂, ^... , D_N respectively such that

I(f_i, f_i) = (|f_i|² - j(u)f_i²) < 0.

Denote W the linear subspace spanned by f₁, f₂, ^... , f_N. Since they have disjoint supports, they are orthogonal and thus the dimension of W is N. The index form I( ^. , ^. ) is negative definite on W so the Morse index is greater than or equal to N.

PROPOSITION 2.2. Let (M, g) be Riemannian manifold and u ³ 0 be a continuous function satisfying the following inequality of Simons' type in the distribution sense

(2.2)

where a > 0 is a constant and is a continuous function on R. If u has a relatively compact nodal domain D, then there exists a function f_D with support in D such that

(|f|² - j(u)f²) < 0.

PROOF. Suppose that u admits a relatively compact nodal domain D. Write q : = (u) and u: = log u on D. Thus (2.2) can be written as

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

(|f|² - qf²) £ - a

f²|u|² + |f - fu|².

Let f =u, for some function to be determined. We obtain

(|f|² - qf²) £ - a

²|u|² + u²||².

For all b such that U/2b

U, where U : = u, set

_b(x) =

Denote D₊ (resp. D_ ) the set of points in D with u(x) ³ b (resp. u(x)b). A simple calculation leads to:

(|f|² - qf²)

u²|u|² - |u|².

When b goes to U, the first term of right hand side tends to 0 (because |u|² is integrable), while the second term is fixed. It follows that (|f|² - qf²) < 0 for all functions f = _bu, when b is close to U. The conclusion is proved.

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• DO CARMO MP AND ZHOU D. 2000. Bernstein-type Theorems in Hypersurfaces with Constant Mean Curvature. An Acad Bras Cienc 72: 301-310.

Publication Dates

• Publication in this collection
  08 Oct 2001
• Date of issue
  Sept 2001

History

• Received
  24 July 2001
• Accepted
  01 Aug 2001