Dear Sirs,
I would like to certificate that Mr Nikolaos Katzourakis was an
undergraduate student (Academic year 2003-2004) when the submitted
paper was completed, titled as "The Riemann Surface of the Logarithm
Constructed in a Geometrical Framework", as justified form the
aknowledgement .
Sincerely yours,
Dr. Anastasios Kartsaklis
University of Athens
Faculty of Sciences
Department of Mathematics
Division of Algebra and Geometry
Dear Prof. Lautzenheiser,
In this paper the author uses pure differential geometric methods to
construct a realization of the Riemann surface of the logarithm in
Euclidean 3-space. The interesting fact is that this is achieved
without employing the working philosophy of any of the related standard
tools of complex manifolds (holomorphic structure and analytic
continuation).
It is well-known that Riemann surfaces considered as complex 1manifolds (of 1 complex dimension) live into Euclidean 4-space (up to
an analytic embedding) but are not embeddable into 3-space.
Notwithstanding, the author sets up a sharp construction via uniform
convergence of smooth helicoid surface to provide finite-many sheets of
the Riemann surface as the limit of the helicoid sequence which is, in
fact, a covering space over the complex plane. This realization is
dimensionwise optimal and in addition admits generalization on
multidiscs and multihelicoid-like submanifolds of the complex n-space,
provided in the last section.
In the light of the above, I do recommend the publication of this
paper.
My best regards,
Anastasios Kartsaklis
Lecturer (BSc Phys-PhD math)
University of Athens
Fuculty of Sciences
Dept. of Mathematics
Section: Algebra an Geometry