Submanifolds of wraped product manifolds I ×f Sm-1(k) from a p-harmonic viewpoint

Abstract

We study p-harmonic maps, p-harmonic morphisms, biharmonic maps, and quasiregular mappings into submanifolds of warped product Riemannian manifolds I×_fS ^m−1(k) of an open interval and a complete simply-connecteded (m − 1)-dimensional Riemannian manifold of constant sectional curvature k. We establish an existence theorem for p-harmonic maps and give a classification of complete stable minimal surfaces in certain three dimensional warped product Riemannian manifolds R ×_f S ²(k), building on our previous work. When f ≡ Const. and k = 0, we recapture a generalized Bernstein Theorem and hence the Classical Bernstein Theorem in R³ . We then extend the classification to parabolic stable minimal hypersurfaces in higher dimensions.