Submanifolds of warped product manifolds $I{\times }_f{S}^{m-1}(k)$}} 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 $I{\times }_f{S}^{m-1}(k)$ 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 $\mathbf{R}{\times }_f{S}^2(k),$ building on our previous work. When $f\equiv$ Const. and $k=0$, we recapture a generalized Bernstein Theorem and hence the Classical Bernstein Theorem in $R^3$. We then extend the classification to parabolic stable minimal hypersurfaces in higher dimensions.

arXiv comment20 pages. Proceeding of the International Conference RIGA-2008, Brasov, Romania, July 8-11, 2008