Submanifolds of warped product manifolds $I\times_f S^{m-1}(k)$}} from a $p$-harmonic viewpoint

TL;DR

This paper investigates various harmonic and quasiregular mappings into submanifolds of warped product manifolds, establishing existence, classification, and extending classical theorems to higher dimensions.

Contribution

It introduces new existence results for p-harmonic maps and classifies stable minimal surfaces and hypersurfaces in warped product manifolds, extending classical theorems.

Findings

Existence theorem for p-harmonic maps into warped products

Classification of complete stable minimal surfaces in 3D warped products

Extension of Bernstein theorem to higher dimensions

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 ${\bf R}\times_f S^{2}(k)\, ,$ 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.