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 R n+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 R n+1 obtained by perturbing noncompact portions of the catenoids. We also consider the moduli space M r,k (g) of elliptic r-minimal hypersurfaces with k ≥ 2 ends of planar type in R n+1 endowed with an ALE metric g, and show that M r,k (g) is an analytic manifold of formal dimension k(n + 1), with M r,k (g) being smooth for a generic g in a neighborhood of the standard Euclidean metric.

Poincaré type inequality for hypersurfaces and rigidity results

“…We recall that a hypersurface M of M m+1 is called r-minimal if H r vanishes on M . Properties of hypersurfaces involving the r-mean curvaure, including the case of r-minimal hypersurfaces, has been object of research by many authors as, for example, [34], [41], [30], [5], [7], [36], [49], and [14]. Associated to the family of higher-order mean curvatures we have the Newton transformations P r : T M → T M , r ∈ {0, .…”