Exterior Monge- Amp`ere Solutions
نویسنده
چکیده
We discuss the Siciak-Zaharjuta extremal function of a real convex body in C n , a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphic disks in projective space passing through prescribed points at infinity. The extremal curves are all complex quadric curves, and the geometry of such curves allows for the determination of the leaves of the foliation by simple geometric criteria. As a byproduct we encounter a new invariant of an exterior domain, the Robin indicatrix, which is in some cases the dual of the Kobayashi indicatrix for a bounded domain.
منابع مشابه
The Obstacle Problem for Monge - Amp Ere Equation
We consider the obstacle problem for the degenerated Monge-Amp ere equation. We prove the existence of the greatest viscosity sub-solution,and the C 1;1-regularity. Then the solution satisses the concave uniformly elliptic equation. We use the author's previous work to show the C 1;;-regularity of the free boundary. Finally, we discuss the stability of the free boundary. In this paper, we consi...
متن کاملExistence and multiplicity of positive solutions for singular Monge-Amp$rmgrave{e}$re system
Using the fixed point theorem in a cone, the existence and multiplicity of radial convex solutions of singular system of Monge-Amp`{e}re equations are established.
متن کاملSpecial Classes of Three Dimensional Affine Hyperspheres Characterized by Properties of Their Cubic Form
It is well known that locally strongly convex a ne hyperspheres can be determined as solutions of di erential equations of Monge-Amp ere type. The global properties of those solutions are well understood. However, due to the nature of the Monge-Amp ere equation, not much is known about local solutions, particularly if the dimension of the hypersurface is greater then 2. By the fundamental theor...
متن کاملEinstein Type Metrics and Stability on Vector Bundles
In this paper we show that stability for holomorphic vector bundles are equivalent to the existence of solutions to certain system of Monge Amp ere equations parametrized by a parameter k. We solve this fully nonlinear elliptic system by singular perturbation technique and show that the vanishing of obstructions for the perturbation is given precisely by the stability condition. This can be int...
متن کامل