p-Harmonic geometry and related topics

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.