arXiv subjects
Publications and source records attributed to Jürgen Leiterer.
Let $\D$ be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map $A:\overline D\to \mathrm{SL}(2,\C)$, which is holomorphic in $\D$, there exist continuous maps $E,F:\overline \D\to \mathfrak{sl}(2,\C)$, which are holomorphic in $\D$, such that $A=e^Ee^F$. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.
We discuss various analytical and geometrical aspects of the Levi form, which is associated with a CR manifold having any CR dimension and any CR codimension.
R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.
R. Guralnick (Linear Algebra Appl. 99, 85-96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, non-smooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global $\mathcal C^\infty$ similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.
R. Guralnick (Linear Algebra Appl. 99, 85-96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, non-smooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global $\mathcal C^\infty$ similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.
Let $X$ be an open subset of $\Bbb C^N$, and let $A$ be an $n\times n$ matrix of holomorphic functions on $X$. We call a point $ξ\in X$ $\mathbf{Jordan}$ $\mathbf{stable}$ for $A$ if $ξ$ is not a splitting point of the eigenvalues of $A$ and, moreover, there is a neighborhood $U$ of $ξ$ such that, for each $1\le k\le n$, the number of Jordan blocks of size $k$ in the Jordan normal forms of $A(ζ)$ is the same for all $ζ\in U$. H. Baumgärtel (Analytic perturbation theory for matrices and operators, Birkhäuser, 1985) proved that there is a nowhere dense closed analytic subset of $X$, which contains all points of $X$ which are not Jordan stable for $A$. We give a new proof of this result. This proof has the advantage that the result can be obtained in a more precise form, and with some estimates. Also, this proof applies to arbitrary, possibly non-smooth, complex spaces $X$.
Using functional analysis and a Friedrichs approximation lemma for first order differential operators, we derive a global homotopy formula in large degrees for the tangential Cauchy-Riemann operator from local homotopy formulas without loss of regularity.