arXiv subjects
Publications and source records attributed to Pascal J. Thomas.
The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.
We prove that a $C^{1,1}$-smooth bounded domain $D$ in $\C^n$ is linearly convex if and only if the convex hull of any two discs in $D$ with common center lies in $D.$
Let $S_ε$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$ε$ tends to 0. To each set $S_ε$ we associate its vanishing ideal $I_ε$ and the pluricomplex Green function $G_ε$ with poles on the set. Suppose that, as $ε$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_ε$ converges to $G$, locally uniformly away from the origin; then the length (i.e. codimension) of $I$ is equal to $N$ and $G \ge G_I$. If the Hilbert-Samuel multiplicity of $I$ is strictly larger than $N$, then $G_ε$ cannot converge to $G_I$. Conversely, if the Hilbert-Samuel multiplicity of $I$ is equal to $N$, (we say that $I$ is a complete intersection ideal), then $G_ε$ does converge to $G_I$. We work out the case of three poles; when the directions defined by any two of the three points converge to limits which don't all coincide, there is convergence, but $G > G_I$.
We prove that an open set $D$ in $\C^n$ is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.
We find the precise growth of some invariant metrics near a point on the boundary of a domain where the Levi form has at least one negative eigenvalue. We also introduce a new invariant pseudometric which is convenient in this context, and give some of its general properties.
The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality is always strict, and the difference between the two functions depends on the order of nilpotence of the strictly upper triangular blocks that appear in the Jordan decomposition of the pole. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carathéodory distance and the Green function, that show that they are distinct in dimension greater or equal to $3$.
The Nevanlinna-Pick problem and the simplest case of the Carathéodory-Fejér problem on the spectral ball $\Om_3$ are reduced to interpolation problems on the symmetrized three-disc $\G_3.$
We discuss some extremal bases for $\CC$-convex domains.
We find all matrices $A$ from the spectral unit ball $Ω_n$ such that the Lempert function $l_{Ω_n}(A,\cdot)$ is continuous.
We give some further criteria for continuity or discontinuity of the Lempert funtion of the spectral ball $Ω_n$, with respect to one or both of its arguments, in terms of cyclicity the matrices involved.
An upper estimate for the Lempert function of any $C^{1+ε}$-smooth bounded domain in $\Bbb C^n$ is found in terms of the boundary distance.
Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.
Necessary and sufficient conditions for Lipschitzness of the Lempert and Green functions are found in terms of their boundary behaviors.
Given a domain $Ω\subset \mathbb C$, the Lempert function is a functional on the space $Hol (\D,Ω)$ of analytic disks with values in $Ω$, depending on a set of poles in $Ω$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.
If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.
Given $A\inΩ_n,$ the $n^2$-dimensional spectral unit ball, we show that $B$ is a "generalized" tangent vector at $A$ to an entire curve in $Ω_n$ if and only if $B$ is in the tangent cone $C_A$ to the isospectral variety at $A.$ In the case of $Ω_3,$ the zero set of this metric is completely described.
Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given.
The Lempert function for several poles $a_0, ..., a_N$ in a domain $Ω$ of $\mathbb C^n$ is defined at the point $z \in Ω$ as the infimum of $\sum^N_{j=0} \log|ζ_j|$ over all the choices of points $ζ_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $Ω$ sending 0 to $z$. 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 $(3/2) \log |z|$, except along three exceptional directions, where it decreases like $2 \log |z|$. The (possible) limit of the corresponding Green functions is not known, and this gives an upper bound for it.