New r-Minimal Hypersurfaces via Perturbative Methods

Abstract

A hypersurface in a Riemannian manifold is r-minimal if its (r+1)-curvature, the (r+1)th elementary symmetric function of its principal curvatures, vanishes identically. If n>2(r+1) we show that the rotationally invariant r-minimal hypersurfaces in ℝⁿ⁺¹ are nondegenerate in the sense that they carry no nontrivial Jacobi fields decaying rapidly enough at infinity. Combining this with a computation of the (r+1)-curvature of normal graphs and the deformation theory in weighted Hölder spaces developed by Mazzeo, Pacard, Pollack, Uhlenbeck and others, we produce new infinite dimensional families of r-minimal hypersurfaces in ℝⁿ⁺¹ obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space ${\mathcal{M}}_{r,k}(g)$ of elliptic r-minimal hypersurfaces with k≥2 ends of planar type in ℝⁿ⁺¹ endowed with an ALE metric g, and show that ${\mathcal{M}}_{r,k}(g)$ is an analytic manifold of formal dimension k(n+1), with ${\mathcal{M}}_{r,k}(g)$ being smooth for a generic g in a neighborhood of the standard Euclidean metric.