18,302 research outputs found
An example of limit of Lempert Functions
The Lempert function for several poles in a domain
of is defined at the point as the infimum of
over all the choices of points in the
unit disk so that one can find a holomorphic mapping from the disk to the
domain sending 0 to . This is always larger than the pluricomplex
Green function for the same set of poles, and in general different.
Here we look at the asymptotic behavior of the Lempert function for three
poles in the bidisk (the origin and one on each axis) as they all tend to the
origin. The limit of the Lempert functions (if it exists) exhibits the
following behavior: along all complex lines going through the origin, it
decreases like , except along three exceptional directions,
where it decreases like . The (possible) limit of the corresponding
Green functions is not known, and this gives an upper bound for it.Comment: 16 pages; references added to related work of the autho
Rigid characterizations of pseudoconvex domains
We prove that an open set in \C^n is pseudoconvex if and only if for
any the largest balanced domain centered at and contained in
is pseudoconvex, and consider analogues of that characterization in the
linearly convex case.Comment: v2: Proposition 14 is improved; v3: Example 15 and the proof of
Proposition 14 are change
On the zero set of the Kobayashi--Royden pseudometric of the spectral unit ball
Given the -dimensional spectral unit ball, we show that
is a "generalized" tangent vector at to an entire curve in
if and only if is in the tangent cone to the isospectral variety at
In the case of the zero set of this metric is completely
described.Comment: minor changes; to appear in Ann. Polon. Mat
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