On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

Explicit formulas for C1,1 and Cconv1,ω extensions of 1-jets in Hilbert and superreflexive spaces

Given X a Hilbert space, ω a modulus of continuity, E an arbitrary subset of X, and functions $f:E\to \mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F,\mathrm{?}F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1,\omega }(X)$, by means of a simple explicit formula. As a consequence of this result, if ω is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes $C_{\mathrm{conv}}^{1,\alpha }(X)$.     

On the Gromov non-hyperbolicity of certain domains in Cn

In this paper, we prove that if Ω is a bounded convex domain in ${\mathbb{C}}^n$, $n\ge 2$, and S is an affine complex hyperplane such that $\mathrm{\Omega }\cap S$ is not empty, then $\mathrm{\Omega }?S$ is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in ${\mathbb{C}}^n$, then $\mathrm{\Omega }=\{(z,w)\in X\times {\mathbb{C}}^{?},\left|w\right|<{\mathrm{e}}^{?\phi (z)}\}$ is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to $\stackrel{￣}{X}$.     

A class of nonpositively curved Kähler manifolds biholomorphic to the unit ball in Cn

Let $(M,g)$ be a simply connected complete Kähler manifold with nonpositive sectional curvature. Assume that g has constant negative holomorphic sectional curvature outside a compact set. We prove that M is then biholomorphic to the unit ball in ${\mathbb{C}}^n$, where ${\dim }_{\mathbb{C}}M=n$. To cite this article: H. Seshadri, K. Verma, C. R. Acad. Sci. Paris, Ser. I 342 (2006).