Vincent Bérard | Searcharxiv

arXiv2012

Exemples de sous variétés de $\mathbb{S}^n$; les variétés proprement biharmoniques avec indice et les variétés proprement C--harmoniques

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some pseudo-umbilical and non pseudo-umbilical examples and we study the link between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the proper conformal-harmonic submanifolds of $\mathbb{S}^n$.

math.DG↗

arXiv2012

Les applications conforme-harmoniques

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 $g_+$.

math.DG↗