# p-Harmonic geometry and related topics

## Keywords:

p-harmonic map, homotopy class, Warped product, minimal submanifold, stable minimal submanifold## Abstract

This paper is both an expository and a research paper on p-harmonic geometry, with a number of examples of p-harmonic maps and some proofs of examples. Rather than giving a comprehensive survey on the subject and citing complete references here, we emphasize topics or problems that do not seem approachable by ordinary harmonic maps (in which *p = 2*). These, in particular include representing homotopy groups by p-harmonic maps, a *p*-harmonic approach to the generalized Bernstein problem, variational problems and PDEs from the p-harmonic viewpoint, main estimates on the function growth in p-harmonic geometry, biharmonic maps and Chen Conjecture on biharmonic immersions, quasiconformal and quasiregular mappings, *1*-harmonic functions, generalized *1*-harmonic equations and the inverse mean curvature flow, and *F*-harmonic maps. Notions of p-balanced and p-imbalanced function growth and volume growth are introduced. Characterizations of entire solutions of constant 1-tension field equation, div ∇f /∇f| = c are given with applications in geometry via transformation group theory. A Liouville Theorem for *F*-harmonic maps is established. We also prove that an immersion *u* of a complete* n*-manifold *M* in a complete manifold *N* is n-energy minimizing (not necessarily conformal) if and only if *u* is conformal and *u(M)* is area-minimizing in *N*.