Polynômes semi-conformes et morphismes harmoniques

We prove a Liouville type theorem for harmonic morphisms from to , showing that any such mapping which is defined off a polar set must be polynomial. We show that any semi-conformal mapping from to defined by polynomials is necessarily harmonic. This result has consequences for the local behaviour of a semi-conformal mapping between arbitrary Riemannian manifolds about a singular point.

---

---

Received December 23, 1997; in final form May 17, 1998     

Distributions à valeurs dans le dual d'un espace de Banach

Sans résumé     

Courants positifs à supports dans une bande

Let T be a positive current of bidimension $(p,p)$ on ${\mathbb{C}}^n$ with support in a strip. If T is closed, S. Giret has proved that T admits a well defined lifting through a blow up with smooth center. The class of positive closed currents with bidimension $(1,1)$ in the unit bidisc ${\Delta }^2$ and with support in a strip, plays a central role in the study of the dynamics of some holomorphic maps. In this note, we prove some estimates of the trace measure of T when $d{d}^{c}T?0$, we prove in particular that if T is closed, then it is algebraic. We then prove two support theorems; the first one in the case where the degree of T is finite and the second in the case where T is positive closed and with tubular support. The latter result generalizes the case $p=n?1$ proved by M. Blel, S.K. Mimouni and G. Raby, which is also a generalization of the case when T is the current of integration on an hypersurface as proved by M.T. Togni. To cite this article: F. Elkhadhra, S.K. Mimouni, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Sur une formule utile dans la détermination de certaines valeurs asymptotiques

Sans résumé     

Note sur les involutions cycliques appartenant à une surface algébrique

Sans résumé     

Triangles inscrits et circonscrits à une courbe convexe sphérique

Sans résumé Présenté par G. Hajós     

Régularité dans une équation de Schrödinger avec potentiel singulier à distance finie et à l'infini

In this Note we study the Schrödinger equation i?_tu+Δu+V₀u+V₁u=0 on $\text{R}{}^3\text{×(0,T)}$ with initial condition $\text{u}{}_0\text{∈{v∈H}{}^2\text{(}\text{R}{}^3\text{)}$, where V₀ is a coulombian potential, singular at finite distance and V₁ is an electric potential, possibly unbounded. Both of them may depend on space and time variables. We prove that this problem is well-posed and that the regularity of the initial data is conserved for the solution. The detailed proof will be given elsewhere (Baudouin et al., in press). To cite this article: L. Baudouin et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Recherche des singularités qui ont rapport à une droite multiple d'une surface

Sans résumé     

Des surfaces à courbure négative constante dans l'espace à trois dimensions et de leurs singularités

Sans résumé