2012/03/25 by Vincent Bérard, Bérard, Vincent
Mathematics · Computer Science · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1203.5512
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 (Mn+1,g+) is an absolute invariant of g+.