On établit une minoration pour la première valeur propre non nulle du problème de Neumann sur les variétés riemanniennes à bord; la nécessité des bornes géométriques utilisées est illustrée par une série d’exemples. Cette approche prolonge celle de Li-Yau, qui était limitée à l’étude du cas où le bord est convexe.
Soit une variété hyperbolique compacte de dimension 3, de diamètre et de volume . Si on note la -ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de , on montre que et , où est une constante ne dépendant que de , et est le nombre de composantes connexes de la partie mince de . En outre, on montre que pour toute 3-variété hyperbolique de volume fini avec cusps, il existe une suite de remplissages compacts de , de diamètre telle que et .
We deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. First, we study some properties of the curvature of mixed 3-Sasakian structures. Then we prove the identity between the class of mixed 3-Sasakian structures and the class of mixed metric 3-contact structures.
We show that a modular class arises from the existence of two generating operators for a Batalin-Vilkovisky algebra. In particular, for every triangular Lie bialgebroid (A,P) such that its top exterior power is a trivial line bundle, there is a section of the vector bundle A whose -cohomology class is well-defined. We give simple proofs of its properties. The modular class of an orientable Poisson manifold is an example. We analyse the relationships between generating operators of the Gerstenhaber...
In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?...
Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....