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Smooth Four Manifolds And Complex Surfaces

Author: Robert Friedman
Publisher: Springer Science & Business Media
ISBN: 3662030284
Size: 25.38 MB
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In 1961 Smale established the generalized Poincare Conjecture in dimensions greater than or equal to 5 [129] and proceeded to prove the h-cobordism theorem [130]. This result inaugurated a major effort to classify all possible smooth and topological structures on manifolds of dimension at least 5. By the mid 1970's the main outlines of this theory were complete, and explicit answers (especially concerning simply connected manifolds) as well as general qualitative results had been obtained. As an example of such a qualitative result, a closed, simply connected manifold of dimension 2: 5 is determined up to finitely many diffeomorphism possibilities by its homotopy type and its Pontrjagin classes. There are similar results for self-diffeomorphisms, which, at least in the simply connected case, say that the group of self-diffeomorphisms of a closed manifold M of dimension at least 5 is commensurate with an arithmetic subgroup of the linear algebraic group of all automorphisms of its so-called rational minimal model which preserve the Pontrjagin classes [131]. Once the high dimensional theory was in good shape, attention shifted to the remaining, and seemingly exceptional, dimensions 3 and 4. The theory behind the results for manifolds of dimension at least 5 does not carryover to manifolds of these low dimensions, essentially because there is no longer enough room to maneuver. Thus new ideas are necessary to study manifolds of these "low" dimensions.

Cartesian Currents In The Calculus Of Variations Ii

Author: Mariano Giaquinta
Publisher: Springer Science & Business Media
ISBN: 9783540640103
Size: 79.37 MB
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This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph

Optical Characterization Of Epitaxial Semiconductor Layers

Author: Günther Bauer
Publisher: Springer Science & Business Media
ISBN: 3642796788
Size: 72.35 MB
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The characterization of epitaxial layers and their surfaces has benefitted a lot from the enormous progress of optical analysis techniques during the last decade. In particular, the dramatic improvement of the structural quality of semiconductor epilayers and heterostructures results to a great deal from the level of sophistication achieved with such analysis techniques. First of all, optical techniques are nondestructive and their sensitivity has been improved to such an extent that nowadays the epilayer analysis can be performed on layers with thicknesses on the atomic scale. Furthermore, the spatial and temporal resolution have been pushed to such limits that real time observation of surface processes during epitaxial growth is possible with techniques like reflectance difference spectroscopy. Of course, optical spectroscopies complement techniques based on the inter action of electrons with matter, but whereas the latter usually require high or ultrahigh vacuum conditions, the former ones can be applied in different environments as well. This advantage could turn out extremely important for a rather technological point of view, i.e. for the surveillance of modern semiconductor processes. Despite the large potential of techniques based on the interaction of electromagnetic waves with surfaces and epilayers, optical techniques are apparently moving only slowly into this area of technology. One reason for this might be that some prejudices still exist regarding their sensitivity.

Stein Manifolds And Holomorphic Mappings

Author: Franc Forstnerič
Publisher: Springer
ISBN: 3319610589
Size: 58.45 MB
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This book, now in a carefully revised second edition, provides an up-to-date account of Oka theory, including the classical Oka-Grauert theory and the wide array of applications to the geometry of Stein manifolds. Oka theory is the field of complex analysis dealing with global problems on Stein manifolds which admit analytic solutions in the absence of topological obstructions. The exposition in the present volume focuses on the notion of an Oka manifold introduced by the author in 2009. It explores connections with elliptic complex geometry initiated by Gromov in 1989, with the Andersén-Lempert theory of holomorphic automorphisms of complex Euclidean spaces and of Stein manifolds with the density property, and with topological methods such as homotopy theory and the Seiberg-Witten theory. Researchers and graduate students interested in the homotopy principle in complex analysis will find this book particularly useful. It is currently the only work that offers a comprehensive introduction to both the Oka theory and the theory of holomorphic automorphisms of complex Euclidean spaces and of other complex manifolds with large automorphism groups./div

Complex Compact Surfaces

Author: W. Barth
Publisher: Springer Science & Business Media
ISBN: 9783540008323
Size: 59.76 MB
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The first edition of "Compact Complex Surfaces" was published in 1984 and has become one of the most important books on the subject. In this second enlarged edition the major developments of the last 20 years have been incorporated. The Enriques-Kodaira classification is carried out in the spirit of Mori theory and many new developments have been added, including new analytic tools as well as new algebraic methods such as the theorems of Bogomolov and Reider and their applications. A new section is devoted to the stunning results achieved by the introduction of Donaldson and Seiberg-Witten invariants.

The Geometry Of Infinite Dimensional Groups

Author: Boris Khesin
Publisher: Springer Science & Business Media
ISBN: 3540772634
Size: 76.55 MB
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This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Geometric Invariant Theory

Author: David Mumford
Publisher: Springer Science & Business Media
ISBN: 9783540569633
Size: 72.63 MB
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"Geometric Invariant Theory" by Mumford/Fogarty (the firstedition was published in 1965, a second, enlarged editonappeared in 1982) is the standard reference on applicationsof invariant theory to the construction of moduli spaces.This third, revised edition has been long awaited for by themathematical community. It is now appearing in a completelyupdated and enlarged version with an additional chapter onthe moment map by Prof. Frances Kirwan (Oxford) and a fullyupdated bibliography of work in this area.The book deals firstly with actions of algebraic groups onalgebraic varieties, separating orbits by invariants andconstructionquotient spaces; and secondly with applicationsof this theory to the construction of moduli spaces.It is a systematic exposition of the geometric aspects ofthe classical theory of polynomial invariants.

Quasi Projective Moduli For Polarized Manifolds

Author: Eckart Viehweg
Publisher: Springer Science & Business Media
ISBN: 3642797458
Size: 24.49 MB
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The concept of moduli goes back to B. Riemann, who shows in [68] that the isomorphism class of a Riemann surface of genus 9 ~ 2 depends on 3g - 3 parameters, which he proposes to name "moduli". A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see [59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In variant Theory". We will recall the necessary tools from his book [59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in [78], a careful study of positivity proper ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf.