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One of the most elementary questions in mathematics is whether an
area minimizing surface spanning a contour in three space is
immersed or not; i.e. does its derivative have maximal rank
everywhere. The purpose of this monograph is to present an
elementary proof of this very fundamental and beautiful
mathematical result. The exposition follows the original line of
attack initiated by Jesse Douglas in his Fields medal work in 1931,
namely use Dirichlet's energy as opposed to area. Remarkably, the
author shows how to calculate arbitrarily high orders of
derivatives of Dirichlet's energy defined on the infinite
dimensional manifold of all surfaces spanning a contour, breaking
new ground in the Calculus of Variations, where normally only the
second derivative or variation is calculated. The monograph begins
with easy examples leading to a proof in a large number of cases
that can be presented in a graduate course in either manifolds or
complex analysis. Thus this monograph requires only the most basic
knowledge of analysis, complex analysis and topology and can
therefore be read by almost anyone with a basic graduate education.
Many properties of minimal surfaces are of a global nature, and
this is already true for the results treated in the first two
volumes of the treatise. Part I of the present book can be viewed
as an extension of these results. For instance, the first two
chapters deal with existence, regularity and uniqueness theorems
for minimal surfaces with partially free boundaries. Here one of
the main features is the possibility of "edge-crawling" along free
parts of the boundary. The third chapter deals with a priori
estimates for minimal surfaces in higher dimensions and for
minimizers of singular integrals related to the area functional. In
particular, far reaching Bernstein theorems are derived. The second
part of the book contains what one might justly call a "global
theory of minimal surfaces" as envisioned by Smale. First, the
Douglas problem is treated anew by using Teichmuller theory.
Secondly, various index theorems for minimal theorems are derived,
and their consequences for the space of solutions to Plateaus
problem are discussed. Finally, a topological approach to minimal
surfaces via Fredholm vector fields in the spirit of Smale is
presented.
Regularity of Minimal Surfaces begins with a survey of minimal
surfaces with free boundaries. Following this, the basic results
concerning the boundary behaviour of minimal surfaces and
H-surfaces with fixed or free boundaries are studied. In
particular, the asymptotic expansions at interior and boundary
branch points are derived, leading to general Gauss-Bonnet
formulas. Furthermore, gradient estimates and asymptotic expansions
for minimal surfaces with only piecewise smooth boundaries are
obtained. One of the main features of free boundary value problems
for minimal surfaces is that, for principal reasons, it is
impossible to derive a priori estimates. Therefore regularity
proofs for non-minimizers have to be based on indirect reasoning
using monotonicity formulas. This is followed by a long chapter
discussing geometric properties of minimal and H-surfaces such as
enclosure theorems and isoperimetric inequalities, leading to the
discussion of obstacle problems and of Plateaus problem for
H-surfaces in a Riemannian manifold. A natural generalization of
the isoperimetric problem is the so-called thread problem, dealing
with minimal surfaces whose boundary consists of a fixed arc of
given length. Existence and regularity of solutions are discussed.
The final chapter on branch points presents a new approach to the
theorem that area minimizing solutions of Plateaus problem have no
interior branch points.
This book consists almost entirely of papers delivered at the
Seminar on partial differential equations held at
Max-Planck-Institut in the spring of 1984. They give an insight
into important recent research activities. Some further
developments are also included.
Regularity of Minimal Surfaces begins with a survey of minimal
surfaces with free boundaries. Following this, the basic results
concerning the boundary behaviour of minimal surfaces and
H-surfaces with fixed or free boundaries are studied. In
particular, the asymptotic expansions at interior and boundary
branch points are derived, leading to general Gauss-Bonnet
formulas. Furthermore, gradient estimates and asymptotic expansions
for minimal surfaces with only piecewise smooth boundaries are
obtained. One of the main features of free boundary value problems
for minimal surfaces is that, for principal reasons, it is
impossible to derive a priori estimates. Therefore regularity
proofs for non-minimizers have to be based on indirect reasoning
using monotonicity formulas. This is followed by a long chapter
discussing geometric properties of minimal and H-surfaces such as
enclosure theorems and isoperimetric inequalities, leading to the
discussion of obstacle problems and of Plateaus problem for
H-surfaces in a Riemannian manifold. A natural generalization of
the isoperimetric problem is the so-called thread problem, dealing
with minimal surfaces whose boundary consists of a fixed arc of
given length. Existence and regularity of solutions are discussed.
The final chapter on branch points presents a new approach to the
theorem that area minimizing solutions of Plateaus problem have no
interior branch points.
These lecture notes are based on the joint work of the author and
Arthur Fischer on Teichmiiller theory undertaken in the years
1980-1986. Since then many of our colleagues have encouraged us to
publish our approach to the subject in a concise format, easily
accessible to a broad mathematical audience. However, it was the
invitation by the faculty of the ETH Ziirich to deliver the ETH N
achdiplom-Vorlesungen on this material which provided the
opportunity for the author to develop our research papers into a
format suitable for mathematicians with a modest background in
differential geometry. We also hoped it would provide the basis for
a graduate course stressing the application of fundamental ideas in
geometry. For this opportunity the author wishes to thank Eduard
Zehnder and Jiirgen Moser, acting director and director of the
Forschungsinstitut fiir Mathematik at the ETH, Gisbert Wiistholz,
responsible for the Nachdiplom Vorlesungen and the entire ETH
faculty for their support and warm hospitality. This new approach
to Teichmiiller theory presented here was undertaken for two
reasons. First, it was clear that the classical approach, using the
theory of extremal quasi-conformal mappings (in this approach we
completely avoid the use of quasi-conformal maps) was not easily
applicable to the theory of minimal surfaces, a field of interest
of the author over many years. Second, many other active
mathematicians, who at various times needed some Teichmiiller
theory, have found the classical approach inaccessible to them.
One of the most elementary questions in mathematics is whether an
area minimizing surface spanning a contour in three space is
immersed or not; i.e. does its derivative have maximal rank
everywhere. The purpose of this monograph is to present an
elementary proof of this very fundamental and beautiful
mathematical result. The exposition follows the original line of
attack initiated by Jesse Douglas in his Fields medal work in 1931,
namely use Dirichlet's energy as opposed to area. Remarkably, the
author shows how to calculate arbitrarily high orders of
derivatives of Dirichlet's energy defined on the infinite
dimensional manifold of all surfaces spanning a contour, breaking
new ground in the Calculus of Variations, where normally only the
second derivative or variation is calculated. The monograph begins
with easy examples leading to a proof in a large number of cases
that can be presented in a graduate course in either manifolds or
complex analysis. Thus this monograph requires only the most basic
knowledge of analysis, complex analysis and topology and can
therefore be read by almost anyone with a basic graduate education.
This textbook by respected authors helps students foster
computational skills and intuitive understanding with a careful
balance of theory, applications, historical development and
optional materials.
Why does nature prefer some shapes and not others? The variety of
sizes, shapes, and irregularities in nature is endless. Skillfully
integrating striking full-color illustrations, the authors describe
the efforts by scientists and mathematicians since the Renaissance
to identify and describe the principles underlying the shape of
natural forms. But can one set of laws account for both the
symmetry and irregularity as well as the infinite variety of
nature's designs? A complete answer to this question is likely
never to be discovered. Yet, it is fascinating to see how the
search for some simple universal laws down through the ages has
increased our understanding of nature. The Parsimonious Universe
looks at examples from the world around us at a non-mathematical,
non-technical level to show that nature achieves efficiency by
being stingy with the energy it expends.
Basic Multivariable Calculus fills the need for a student-oriented
text devoted exclusively to the third-semester course in
multivariable calculus. In this text, the basic algebraic,
analytic, and geometric concepts of multivariable and vector
calculus are carefully explained, with an emphasis on developing
the student's intuitive understanding and computational technique.
A wealth of figures supports geometrical interpretation, while
exercise sets, review sections, practice exams, and historical
notes keep the students active in, and involved with, the
mathematical ideas. All necessary linear algebra is developed
within the text, and the material can be readily coordinated with
computer laboratories. Basic Multivariable Calculus is the product
of an extensive writing, revising, and class-testing collaboration
by the authors of Calculus III (Springer-Verlag) and Vector
Calculus (W.H. Freeman & Co.). Incorporating many features from
these highly respected texts, it is both a synthesis of the
authors' previous work and a new and original textbook.
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