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Author : Friedrich Tomi
Publisher : Unknown
Page : 78 pages
File Size : 55,8 Mb
Release : 1995
Category : Index theorems
ISBN : 1470401398
Author : Friedrich Tomi,Anthony Tromba
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 55,8 Mb
Release : 1995
Category : Index theorems
ISBN : 9780821803523
In this paper we formulate and prove an index theorem for minimal surfaces of higher topological type spanning one boundary contour. Our techniques carry over to surfaces with several boundary contours as well as to unoriented surfaces.
Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 547 pages
File Size : 45,6 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117060
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 Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau ́s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.
Author : Ulrich Dierkes,Stefan Hildebrandt,Anthony Tromba
Publisher : Springer Science & Business Media
Page : 634 pages
File Size : 46,7 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642117008
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 Plateau ́s 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 Plateau ́s problem have no interior branch points.
Author : Ulrich Dierkes,Stefan Hildebrandt,Albrecht Küster,Ortwin Wohlrab
Publisher : Springer Science & Business Media
Page : 528 pages
File Size : 53,5 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 9783662027912
Minimal surfaces I is an introduction to the field of minimal surfaces and apresentation of the classical theory as well as of parts of the modern development centered around boundary value problems. Part II deals with the boundary behaviour of minimal surfaces. Part I is particularly apt for students who want to enter this interesting area of analysis and differential geometry which during the last 25 years of mathematical research has been very active and productive. Surveys of various subareas will lead the student to the current frontiers of knowledge and can alsobe useful to the researcher. The lecturer can easily base courses of one or two semesters on differential geometry on Vol. 1, as many topics are worked out in great detail. Numerous computer-generated illustrations of old and new minimal surfaces are included to support intuition and imagination. Part 2 leads the reader up to the regularity theory fornonlinear elliptic boundary value problems illustrated by a particular and fascinating topic. There is no comparably comprehensive treatment of the problem of boundary regularity of minimal surfaces available in book form. This long-awaited book is a timely and welcome addition to the mathematical literature.
Author : Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny
Publisher : Springer Science & Business Media
Page : 699 pages
File Size : 44,8 Mb
Release : 2010-08-16
Category : Mathematics
ISBN : 9783642116988
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 nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling ́s 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 Plateau ́s 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 Nitsche ́s uniqueness theorem and Tomi ́s finiteness result. In addition, a theory of unstable solutions of Plateau ́s problems is developed which is based on Courant ́s mountain pass lemma. Furthermore, Dirichlet ́s problem for nonparametric H-surfaces is solved, using the solution of Plateau ́s problem for H-surfaces and the pertinent estimates.
Author : Martin Ulrich Schmidt
Publisher : American Mathematical Soc.
Page : 111 pages
File Size : 51,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804605
This memoir develops the spectral theory of the Lax operators of nonlinear Schrodinger-like partial differential equations with periodic boundary conditions. Their spectral curves, i.e., the common spectrum with the periodic shifts, are generically Riemann surfaces of infinite genus. The points corresponding to infinite energy are added. The resulting spaces are no longer Riemann surfaces in the usual sense, but they are quite similar to compact Riemann surfaces. In fact, some of the basic tools of the theory of compact Riemann surfaces are generalized to these spectral curves and illuminate the structure of complete integrability: The eigen bundles define holomorphic line bundles on the spectral curves, which completely determine the potentials. These line bundles may be described by divisors of the same degree as the genus, and these divisors give rise to Darboux coordinates. With the help of a Riemann-Roch Theorem, the isospectral sets (the sets of all potentials corresponding to the same spectral curve) may be identified with open dense subsets of the Jacobian varieties. The real parts of the isospectral sets are infinite dimensional tori, and the group action solves the corresponding nonlinear partial differential equations. Deformations of the spectral curves are in one to one correspondence with holomorphic forms. Serre Duality reproduces the symplectic form.
Author : James Damon
Publisher : American Mathematical Soc.
Page : 130 pages
File Size : 52,6 Mb
Release : 1996
Category : Holomorphic mappings
ISBN : 9780821804810
Almost free divisors and complete intersections form a general class of nonisolated hypersurface and completer intersection singularities. They also include discriminants of mappings, bifurcation sets, and certain types of arrangements of hyperplanes such as Coxeter arrangements and generic arrangements. Associated to the singularities of this class is a "singular Milnor fibration" which has the same homotopy properties as the Milnor fibration for isolated singularities. This memoir deduces topological properties of singularities in a number of situations including: complements of hyperplane arrangements, various nonisolated complete intersections, nonlinear arrangements of hypersurfaces, functions on discriminants, singularities defined by compositions of functions, and bifurcation sets.
Author : Wayne Aitken
Publisher : American Mathematical Soc.
Page : 174 pages
File Size : 45,5 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804070
The first half of this work gives a treatment of Deligne's functorial intersection theory tailored to the needs of this paper. This treatment is intended to satisfy three requirements: 1) that it be general enough to handle families of singular curves, 2) that it be reasonably self-contained, and 3) that the constructions given be readily adaptable to the process of adding norms and metrics such as is done in the second half of the paper. The second half of the work is devoted to developing a class of intersection functions for singular curves that behaves analogously to the canonical Green's functions introduced by Arakelov for smooth curves. These functions are called intersection functions since they give a measure of intersection over the infinite places of a number field. The intersection over finite places can be defined in terms of the standard apparatus of algebraic geometry. Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem.
Author : Liviu I. Nicolaescu
Publisher : American Mathematical Soc.
Page : 98 pages
File Size : 40,9 Mb
Release : 1997
Category : Geometry, Differential
ISBN : 9780821806210
In this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family. All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eigh symmetries in the real case and two in the complex case). This text will also be of interest to those working in geometry and topology.
Author : Paul Concus,Robert Finn
Publisher : Springer Science & Business Media
Page : 201 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461246565
Vallombrosa Center was host during the week September 7-12, 1985 to about 40 mathematicians, physical scientists, and engineers, who share a common interest in free surface phenomena. This volume includes a selection of contributions by participants and also a few papers by interested scientists who were unable to attend in person. Although a proceedings volume cannot recapture entirely the stimulus of personal interaction that ultimately is the best justification for such a gathering, we do offer what we hope is a representative sampling of the contributions, indicating something of the varied and interrelated ways with which these classical but largely unsettled questions are currently being attacked. For the participants, and also for other specialists, the 23 papers that follow should help to establish and to maintain the new ideas and insights that were presented, as active working tools. Much of the material will certainly be of interest also for a broader audience, as it impinges and overlaps with varying directions of scientific development. On behalf of the organizing committee, we thank the speakers for excellent, well-prepared lectures. Additionally, the many lively informal discussions did much to contribute to the success of the conference.
Author : F.J.E. Dillen,L.C.A. Verstraelen
Publisher : Elsevier
Page : 1067 pages
File Size : 43,8 Mb
Release : 1999-12-16
Category : Mathematics
ISBN : 9780080532837
In the series of volumes which together will constitute the Handbook of Differential Geometry a rather complete survey of the field of differential geometry is given. The different chapters will both deal with the basic material of differential geometry and with research results (old and recent). All chapters are written by experts in the area and contain a large bibliography.
Author : Kazuyoshi Kiyohara
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 53,7 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806401
In this work, two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.
Author : Karl Heinrich Hofmann,Wolfgang Ruppert,Wolfgang A. F. Ruppert
Publisher : American Mathematical Soc.
Page : 189 pages
File Size : 44,5 Mb
Release : 1997
Category : Exponential functions
ISBN : 9780821806418
In the structure theory of real Lie groups, there is still information lacking about the exponential function. Most notably, there are no general necessary and sufficient conditions for the exponential function to be surjective. It is surprising that for subsemigroups of Lie groups, the question of the surjectivity of the exponential function can be answered. Under nature reductions setting aside the "group part" of the problem, subsemigroups of Lie groups with surjective exponential function are completely classified and explicitly constructed in this memoir. There are fewer than one would think and the proofs are harder than one would expect, requiring some innovative twists. The main protagonists on the scene are SL(2, R) and its universal covering group, almost abelian solvable Lie groups (ie. vector groups extended by homotheties), and compact Lie groups. This text will also be of interest to those working in algebra and algebraic geometry.
Author : Michael A. Dritschel,Hugo Jan Woerdeman
Publisher : American Mathematical Soc.
Page : 62 pages
File Size : 47,8 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806517
This memoir initiates a model theory-based study of the numerical radius norm. Guided by the abstract model theory of Jim Agler, the authors propose a decomposition for operators that is particularly useful in understanding their properties with respect to the numerical radius norm. Of the topics amenable to investigation with these tools, the following are presented: A complete description of the linear extreme points of the $n\times n$ matrix (numerical radius) unit ball Several equivalent characterizations of matricial extremals in the unit ball; that is, those members which do not allow a nontrivial extension remaining in the unit ball Applications to numerical ranges of matrices, including a complete parameterization of all matrices whose numerical ranges are closed disks In addition, an explicit construction for unitary 2-dilations of unit ball members is given, Ando's characterization of the unit ball is further developed, and a study of operators satisfying $ A - \mathrm{Re} (e^{i\theta}A)\geq 0$ for all $\theta$ is initiated.
