Exterior Monge–Ampère solutions


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We discuss the Siciak–Zaharjuta extremal function of a real convex body in Cn, 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 quadratic 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. Finally, we construct extremal curves for two non-convex bodies in R2.

Journal: Advances in Mathematics - ADVAN MATH , vol. 222, no. 2, pp. 331-358, 2009

DOI: 10.1016/j.aim.2009.01.008

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References (10)

Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in ${\bf R}^n$ (Citations: 3)

The dirichlet problem for a complex Monge-Ampère equation (Citations: 34)

Pluripotential theory for convex bodies in R N (Citations: 5)

Convex Bodies Associated with a Convex Body (Citations: 11)

Symmetries and other transformations of the complex Monge - Amp`ere equation (Citations: 8)


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