New \(r\)-minimal hypersurfaces via perturbative methods (Q645299)

A hypersurface is called \(r\)-minimal if and only if the \((r+1)\)-th elementary symmetric function of its principal curvatures vanishes identically. Using the deformation theory in weighted Hölder spaces developed by Mazzeo, Pacard, Pollack and Uhlenbeck, the authors produce new infinite-dimensional families of \(r\)-minimal hypersurfaces of the \((n+1)\)-dimensional Euclidean space. These examples are obtained by perturbing noncompact portions of catenoids. The authors also consider the moduli space of elliptic \(r\)-minimal hypersurfaces with \(k \geq 2\) ends of planar type endowed with an ALE metric \(g\). They show that this is an analytic manifold of formal dimension \(k(n+1)\) which is smooth for a generic \(g\) in a neighborhood of the standard Euclidean metric.