4,641 research outputs found

    An example of limit of Lempert Functions

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    The Lempert function for several poles a0,...,aNa_0, ..., a_N in a domain Ω\Omega of Cn\mathbb C^n is defined at the point z∈Ωz \in \Omega as the infimum of ∑j=0Nlog⁥∣ζj∣\sum^N_{j=0} \log|\zeta_j| over all the choices of points ζj\zeta_j in the unit disk so that one can find a holomorphic mapping from the disk to the domain Ω\Omega sending 0 to zz. 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 (3/2)log⁥∣z∣(3/2) \log |z|, except along three exceptional directions, where it decreases like 2log⁥∣z∣2 \log |z|. 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

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    We prove that an open set DD in \C^n is pseudoconvex if and only if for any z∈Dz\in D the largest balanced domain centered at zz and contained in DD 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

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    Given A∈Ωn,A\in\Omega_n, the n2n^2-dimensional spectral unit ball, we show that BB is a "generalized" tangent vector at AA to an entire curve in Ωn\Omega_n if and only if BB is in the tangent cone CAC_A to the isospectral variety at A.A. In the case of Ω3,\Omega_3, the zero set of this metric is completely described.Comment: minor changes; to appear in Ann. Polon. Mat

    "Convex" characterization of linearly convex domains

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    We prove that a C1,1C^{1,1}-smooth bounded domain DD in \C^n is linearly convex if and only if the convex hull of any two discs in DD with common center lies in D.D.Comment: to appear in Math. Scand.; v3: Appendix is adde