Bøger i Grundlehren der mathematischen Wissenschaften serien

Seit den Anfängen der Theorie der Minimalflächen vor mehr als zwei Jahrhunderten sind viele große Geister aller Epochen von ihrem Reize fasziniert worden. Diese Anziehungskraft liegt nicht nur in dem geome­ trischen Gehalt der Theorie und in ihren inspirierenden Einwirkungen auf die Entwicklung mathematischen Gedankenguts begründet, sie er­ klärt sich auch durch die in der Mathematik wohl nur selten erreichte Vielschichtigkeit, mit welcher in ihr sowohl experimenteller Augenschein und die Verfolgung konkreter Einzelprobleme als auch die fortschrei­ tende Abstraktion ursprünglich anschaulicher Begriffe und die Durch­ schlagskraft allgemein anwendbarer Methoden erfolgreich zum Tragen kommen. Es bestehen innige Zusammenhänge mit der lokalen und glo­ balen Differentialgeometrie, mit der Funktionentheorie, der Variations­ rechnung und der Theorie partieller Differentialgleichungen und zugleich fruchtbare Beziehungen zu vielen mathematischen Gebieten, so etwa zur Topologie, zur Maßtheorie und zur algebraischen Geometrie. Auch der Forscher in anderen Disziplinen, beispielsweise in der Elastizitäts­ theorie, der Strömungslehre und in allen Gebieten, bei denen die Er­ scheinung der Kapillarität eine Rolle spielt, wird von seiner Vertrautheit mit Minimalflächen profitieren. Vor allem aber handelt es sich um eine ästhetisch vollkommene Materie. Mit Ausnahme einiger spärlich gehaltenen Andeutungen befaßt sich die vorliegende Monographie ausschließlich mit zweidimensionalen reellen Parameterflächen im dreidimensionalen Euklidischen Raum. Eine solche Begrenzung schien aus Platzgründen und im Hinblick auf den Wunsch nach Stoffeinheitlichkeit unerläßlich. Der Kritiker kann hier freilich jedes Wort als lästige Einschränkung empfinden, wird aber hoffentlichzugestehen, daß die schönsten Perlen der Theorie dennoch in Erscheinung treten.

This book provides a coherent, self-contained introduction to central topics of Analytic Partial Differential Equations in the natural geometric setting. The main themes are the analysis in phase-space of analytic PDEs and the FourierΓÇôBrosΓÇôIagolnitzer (FBI) transform of distributions and hyperfunctions, with application to existence and regularity questions.The book begins by establishing the fundamental properties of analytic partial differential equations, starting with the CauchyΓÇôKovalevskaya theorem, before presenting an integrated overview of the approach to hyperfunctions via analytic functionals, first in Euclidean space and, once the geometric background has been laid out, on analytic manifolds. Further topics include the proof of the Lojaciewicz inequality and the division of distributions by analytic functions, a detailed description of the Frobenius and Nagano foliations, and the HamiltonΓÇôJacobi solutions of involutive systems of eikonal equations. The reader then enters the realm of microlocal analysis, through pseudodifferential calculus, introduced at a basic level, followed by Fourier integral operators, including those with complex phase-functions (├á la Sj├╢strand). This culminates in an in-depth discussion of the existence and regularity of (distribution or hyperfunction) solutions of analytic differential (and later, pseudodifferential) equations of principal type, exemplifying the usefulness of all the concepts and tools previously introduced. The final three chapters touch on the possible extension of the results to systems of over- (or under-) determined systems of these equationsΓÇöa cornucopia of open problems.This book provides a unified presentation of a wealth of material that was previously restricted to research articles. In contrast to existing monographs, the approach of the book is analytic rather than algebraic, and tools such as sheaf cohomology, stratification theory of analytic varieties and symplectic geometry are used sparingly and introduced as required. The first half of the book is mainly pedagogical in intent, accessible to advanced graduate students and postdocs, while the second, more specialized part is intended as a reference for researchers.

The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition.The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen,the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.

This book gives an overview of research in the topology and geometry of intersections of quadrics in $\mathbb{R}^n$, with a focus on intersections of concentric ellipsoids and related spaces. Unifying and organizing material previously spread over many articles, it also contains new results.The first part provides very detailed foundations of a wide-ranging theory that could be useful for future developments. It includes chapters on general intersections of quadrics, operations on them, and intersections of concentric and coaxial quadrics. Moving from the general to the specific, the second part focuses on a topological description of transverse intersections of concentric ellipsoids, including a complete description of the case of three ellipsoids, and of some large families of more than three of them. The third part looks at relations to other areas of mathematics such as dynamical systems, complex geometry, contact and symplectic geometry, andother applications. An appendix gathers some technical items and also gives an account of the origins, motivations and progression of the subject, including historical recollections of the author, who has been central to its development.

This book provides an in depth discussion of Loewner's theorem on the characterization of matrix monotone functions.