Pseudo-Riemannian and Hessian geometry related to Monge-Ampère structures

On Hessian Riemannian Structures

In Proposition 4.1 a characterization is given of Hessian Rieman-nian structures in terms of a natural connection in the general linear group GL(n; R) + , which is viewed as a principal SO(n)-bundle over the space of positive deenite symmetric n n-matrices. For n = 2, Proposition 5.3 contains an interpretation of the curvature of a Hessian Riemannian structure at a given point, in terms of an u...

1 Pseudo - Riemannian Geometry

Generalized pseudo-Riemannian geometry

Generalized tensor analysis in the sense of Colombeau’s construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a “Fun...

Quaternionic Monge-ampère Equations

The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in H. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].

A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds

where M is a compact complete Riemannian manifold, λ is a positive real parameter, and φ 0 (x) : M → R is a smooth function. We show a meaningful precisely asymptotic result which is more general than those in [1]. Monge-Ampère equations arise naturally from some problems in differential geometry. The existence and regularity of solutions to Monge-Ampère equations have been investigated by many...