More Weakly Biharmonic Maps from the Ball to the Sphere.

In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable critical points of the bienergy.

Eigenvalue Estimates on Weighted Manifolds.

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for -minimal hypersurfaces and the spectrum of the Hodge Laplacian.

On the Normal Stability of Triharmonic Hypersurfaces in Space Forms.

This article is concerned with the stability of triharmonic maps and in particular triharmonic hypersurfaces. After deriving a number of general statements on the stability of triharmonic maps we focus on the stability of triharmonic hypersurfaces in space forms, where we pay special attention to their normal stability. We show that triharmonic hypersurfaces of constant mean curvature in Euclidean space are weakly stable with respect to normal variations while triharmonic hypersurfaces of constant mean curvature in hyperbolic space are stable with respect to normal variations.

On -harmonic self-maps of spheres.

In this manuscript we study rotationally -harmonic maps between spheres. We prove that for () given, there exist infinitely many -harmonic self-maps of for each with . In the case of the identity map of we explicitly determine the spectrum of the corresponding Jacobi operator and show that for , the identity map of is equivariantly stable when interpreted as a -harmonic self-map of .

Polyharmonic hypersurfaces into pseudo-Riemannian space forms.

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form of dimension and index ( and ). We shall study hypersurfaces which are polyharmonic of order (briefly, -harmonic), where and either or . Let denote the shape operator of .