We complete the global classification of spacelike surfaces in the Minkowski three-space with constant Gaussian curvature in terms of harmonic diffeomorphisms onto the hyperbolic plane. A harmonic representation of them is also obtained.

In our preceding articles, by applying the results of Bieberbach, Fatou and Picard, we have constructed an example such that the domain of holomorphy of the solution admits an exterior point. Next we have studied the domain of holomorphy of the solution for differential operators with polynomial coefficients that concern the differential equations of Darboux-Halphen, of Chazy and of the modular function. In this article, we make some studies on the analytic continuation of the solution of the Cauchy problem for certain differential operators of principal part with polynomial coefficients.

Dans nos articles précédents, en appliquant les résultats de Bieberbach, Fatou et Picard, nous avons construit un exemple tel que le domaine d'holomorphie de la solution ademette un point extérieur. Puis nous avons étudié le domaine d'holomorphie de la solution pour les opérateurs différentiels à coefficients polynomiaux, qui concernent les équations différentielles de Darboux-Halphen, de Chazy et de la fonction modulaire. Dans cet article, nous donnons quelques études sur le prolongement analytique de la solution du problème de Cauchy pour certains opérateurs différentiels de partie principale à coefficients polynomiaux.

The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.

We give necessary and sufficient conditions for embeddability of CR manifolds in terms of transversal group actions. These actions do not necessarily preserve the CR structure but they possess a holomorphic extension.

We study the eigenvalues of the biharmonic operators and the buckling eigenvalue on complete, open Riemannian manifolds. We show that the first eigenvalue of the biharmonic operator on a complete, parabolic Riemannian manifold is zero. We give a generalization of the buckling eigenvalue and give applications to studying the stability of minimal Lagrangian submanifolds in Kähler manifolds.

We show that, if g ≥ 2, the virtual cohomological dimension of the mapping class group of a three-dimensional handlebody of genus $\g$ is equal to $4\g-5$ and its Euler number is equal to 0.

The main purpose of this notes is to supplement the paper by Reid: Decomposition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. As an application, a generalization of Fujita's conjecture for singular toric varieties is obtained. We also prove that every toric variety has a small projective toric $\bQ$-factorialization.

We study the blow-up behaviors of solutions of a semilinear heat equation with a nonlinear boundary condition. Under certain conditions, we prove that the blow-up point occurs only at the boundary. Then, by applying the well-known method of Giga-Kohn, we derive the time asymptotic of solutions near the blow-up time. Finally, we prove that the blow-up is complete.

A compact minimal Lagrangian submanifold immersed in a Kähler manifold is called Hamiltonian stable if the second variation of its volume is nonnegative under all Hamiltonian deformations. We study compact Hamiltonian stable minimal Lagrangian submanifolds with parallel second fundamental form embedded in complex projective spaces. Moreover, we completely determine Hamiltonian stability of all real forms in compact irreducible Hermitian symmetric spaces, which were classified previously by M. Takeuchi.