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Minimal Surfaces (Paperback, Softcover reprint of hardcover 2nd ed. 2010)
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Minimal Surfaces (Paperback, Softcover reprint of hardcover 2nd ed. 2010)
Series: Grundlehren der mathematischen Wissenschaften, 339
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Minimal Surfaces is the first volume of a three volume treatise on
minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read
and studied independently of the others. The central theme is
boundary value problems for minimal surfaces. The treatise is a
substantially revised and extended version of the monograph Minimal
Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume
begins with an exposition of basic ideas of the theory of surfaces
in three-dimensional Euclidean space, followed by an introduction
of minimal surfaces as stationary points of area, or equivalently,
as surfaces of zero mean curvature. The final definition of a
minimal surface is that of a non-constant harmonic mapping X:
\Omega\to\R DEGREES3 which is conformally parametrized on
\Omega\subset\R DEGREES2 and may have branch points. Thereafter the
classical theory of minimal surfaces is surveyed, comprising many
examples, a treatment of Bjorlings initial value problem,
reflection principles, a formula of the second variation of area,
the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The
second part of this volume begins with a survey of Plateaus problem
and of some of its modifications. One of the main features is a
new, completely elementary proof of the fact that area A and
Dirichlet integral D have the same infimum in the class C(G) of
admissible surfaces spanning a prescribed contour G. This leads to
a new, simplified solution of the simultaneous problem of
minimizing A and D in C(G), as well as to new proofs of the mapping
theorems of Riemann and Korn-Lichtenstein, and to a new solution of
the simultaneous Douglas problem for A and D where G consists of
several closed components. Then basic facts of stable minimal
surfaces are derived; this is done in the context of stable
H-surfaces (i.e. of stable surfaces of prescribed mean curvature
H), especially of cmc-surfaces (H = const), and leads to curvature
estimates for stable, immersed cmc-surfaces and to Nitsches
uniqueness theorem and Tomis finiteness result. In addition, a
theory of unstable solutions of Plateaus problems is developed
which is based on Courants mountain pass lemma. Furthermore,
Dirichlets problem for nonparametric H-surfaces is solved, using
the solution of Plateaus problem for H-surfaces and the pertinent
estimates."
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