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

Abstract

An erratum to Lemma 2.1 in Do Carmo and Zhou (2000) is presented.Key words: Riemannian manifold, eigenvalue, hypersurface, mean curvature. ERRATUMReplace 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. A RESULT ON NODAL DOMAINSIn this section we prove a result on the nodal domains of |φ| 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.