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<<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 1-manifolds (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
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