Annals of Global Analysis and Geometry 17: 397–407, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 397 Uniqueness and Nonexistence Theorems for Hypersurfaces with H r = 0 JORGE HOUNIE ⋆ Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13.565-905, Brasil. e-mail: hounie@power.ufscar.br MARIA LUIZA LEITE ⋆ Departamento de Estatística, Universidade Federal de Pernambuco, Recife, PE 50.740-540, Brasil Abstract. We use the Maximum Principle to derive theorems of uniqueness and nonexistence for embedded, multiply connected and compact hypersurfaces with boundaries in parallel hyperplanes, which satisfy the geometric property H r = 0. Mathematics Subject Classifications (1991): Primary 53C42; Secondary 53C21. Key words: higher-order curvature functions, maximum principle. 1. Introduction Consider a second-order ordinary differential equation defined on a manifold. The 2-point problem for an ODE is to determine whether two given points can be joined by a curve that satisfies the equation or, in a more precise version, whether they can be joined by exactly k different curves satisfying the ODE, where k = 0, 1, 2,... [8, 10]. A higher-dimensional analogue of this problem is as follows: we have a family S ={} of connected hypersurfaces with boundary that satisfy a second-order PDE on an (n + 1)-dimensional manifold, n ≥ 2, and we ask ourselves whether two given disjoint (n − 1)-dimensional submanifolds D 1 and D 2 can be joined by some  ∈ S in the sense that ∂ = D 1 ∪ D 2 . A typical instance of the first situation concerns geodesic flow while its generalization may be exemplified by the problem of finding a minimal surface spanning a boundary made up of two connected pieces. Soap bubbles are ideally represented by minimal surfaces in R 3 and there is obvious physical evidence that a soap film stretched between a couple of ring shaped pieces of wire necessarily breaks as soon as the rings are taken far enough apart, which suggests that given D 1 and D 2 there is a critical distance bey- ond which no minimal surface may join them. This fact can be rigorously proved ⋆ Both authors were partially supported by CNPq.