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 ƒ has first and second derivatives decaying fast enough at infinity: Its Hessian operator D2 ƒ has at least n − r null eigenvalues everywhere.
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Barroso, C.S., de Lima, L.L. & Santos, W. Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs. J Geom Anal 14, 557–566 (2004). https://doi.org/10.1007/BF02922169
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DOI: https://doi.org/10.1007/BF02922169