Les applications conforme-harmoniques

TL;DR

This paper generalizes the concept of harmonic maps to even-dimensional manifolds, introducing C-harmonic maps as critical points of a conformally invariant functional, and explores their properties and invariants in various geometric contexts.

Contribution

It extends harmonic map theory to even dimensions, constructs a new conformal invariant functional, and analyzes the properties of C-harmonic maps and associated invariants.

Findings

C-harmonic maps satisfy a nonlinear elliptic PDE of order n

The GJMS operator appears as a special case for functions

The asymptotic energy expansion constant is an absolute invariant in AHE manifolds

Abstract

On a Riemannian surface, the energy of a map into a Riemannian manifold is a conformal invariant functional, and its critical points are the harmonic maps. Our main result is a generalization of this theorem when the starting manifold is even dimensional. We then build a conformal invariant functional for the maps between two Riemannian manifolds. Its critical points then called C--harmonic are the solutions of a nonlinear elliptic PDE of order $n$, which is conformal covariant with respect to the start manifold. For the trivial case of real or complex functions of $M$, we find again the GJMS operator, with a leading part power to the $n/2$ of the Laplacian. When $n$ is odd, we prove that the constant term of the asymptotic expansion of the energy of an asymptotically harmonic map on an AHE manifold $(M^{n+1},g_+)$ is an absolute invariant of $g_+$.