The Journal of Geometric Analysis Volume , Number , Monotonicity inequalities for the r -area and a degeneracy theorem for r -minimal graphs Cleon S. Barroso ∗ Levi L. de Lima † Walcy Santos ‡ ABSTRACT. We establish monotonicity inequalities for the r-area of a complete oriented properly immersed r-minimal hypersurface in Euclidean space under appropriate quasi-positivity assumptions on certain invariants of the immersion. The proofs are based on the corresponding first variational formula. As an application, we derive a degeneracy theorem for an entire r-minimal graph whose defining function f has first and second derivatives decaying fast enough at infinity: its Hessian operator D 2 f has at least n − r null eigenvalues everywhere. Key words: r-area, graphs, monotonicity MS Subject Classification: 53A10 - 53C42 1. Introduction Let M ⊂ R n+1 be a smooth orientable hypersurface. Recall that one has a globally defined map N : M → S n , the unit normal Gauss map, well determined up to sign, which fixes an orientation for M and induces for x ∈ M the shape operator A x : T x M → T x M, given by A x (v)= −(D v N )(x), where D is the standard covariant derivative in R n+1 . Since each A x is a symmetric endomorphism of T x M , it has n real eigenvalues, * Department of Mathematics, Federal University of Cear´a, Fortaleza, Brazil. E-mail: cleonbar@mat.ufc.br. † Department of Mathematics, Federal University of Cear´a, Fortaleza, Brazil, partially supported by CNPq and FINEP. E-mail: levi@mat.ufc.br. ‡ Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil, partially supported by CNPq and FAPERJ. E-mail: walcy@im.ufrj.br. c  The Journal of Geometric Analysis ISSN 1050-6926