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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Classification Theory of Riemann Surfaces (Paperback, Softcover reprint of the original 1st ed. 1970)
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Classification Theory of Riemann Surfaces (Paperback, Softcover reprint of the original 1st ed. 1970)
Series: Grundlehren der mathematischen Wissenschaften, 164
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The purpose of the present monograph is to systematically develop a
classification theory of Riemann surfaces. Some first steps will
also be taken toward a classification of Riemannian spaces. Four
phases can be distinguished in the chronological background: the
type problem; general classification; compactifications; and
extension to higher dimensions. The type problem evolved in the
following somewhat overlapping steps: the Riemann mapping theorem,
the classical type problem, and the existence of Green's functions.
The Riemann mapping theorem laid the foundation to classification
theory: there are only two conformal equivalence classes of
(noncompact) simply connected regions. Over half a century of
efforts by leading mathematicians went into giving a rigorous proof
of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE,
HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The
classical type problem was to determine whether a given simply
connected covering surface of the plane is conformally equivalent
to the plane or the disko The problem was in the center of interest
in the thirties and early forties, with AHLFORS, KAKUTANI,
KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMULLER and others
obtaining incisive specific results. The main problem of finding
necessary and sufficient conditions remains, however, unsolved."
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