59 research outputs found
A Maximum Rank Theorem for Solutions to the Homogenous Complex Monge-Amp\`ere Equation in a -Convex Ring
Suppose are two bounded strongly -convex
domains in , with and
. Let
. We call a
-convex ring. We will show that for a solution to the
homogenous complex Monge-Amp\`ere equation in , with on
and on ,
has rank and the level sets of
are strongly -convex
The Preservation of Convexity by Geodesics in the Space of K\"ahler Potentials on Complex Affine Manifolds
On a compact complex affine manifold with a constant coefficient K\"ahler
metric , we introduce a concept: -convexity and show
that -convexity is preserved by geodesics in the space of
K\"ahler potentials. This implies that if two potentials are both strictly
-convex, then the metrics along the geodesic connecting them are
non-degenerate.Comment: 2 figure
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