Sobre hipersuperficies r-minimas com fins planares no espaço euclidiano.

Abstract

A hypersurface Σ ⊂ Rn+1 is r-minimal if its (r + 1) th-curvature (the (r + 1) th elementary symmetric function of its principal curvatures) vanishes identically. If n > 2(r + 1) we showthat the rotationally invariant r-minimal hypersurfaces in R n+1(catenoids) first described in [HL1] are nondegenerate in the sense that they do not carry Jacobi fields which decay rapidly enough at infinity. Combining this with the deformation theory in weighted H¨older spaces developed by Kusner, Mazzeo, Pacard, Pollack, Uhlenbeck and others, we obtain new results on the structure of r-minimal hypersurfaces with ends of planar type. For example, we show that the moduli space Mr,k of complete r-minimal hypersurfaces in Euclidean space R n+1, n > 2(r+1), with k > 2 ends of planar type has the structure of an analytic manifold of virtual dimension k(n+1), which is attained in a neighborhood of a nondegenerate element. Also, we produce new infinite dimensional families of examples of r-minimal hypersurfaces obtained by perturbing noncompact portions of the catenoids. These seem to be the first known families of examples of noncompact elliptic r-minimal hypersurfaces without symmetries.