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The description for this book, Topological Methods in the Theory of Functions of a Complex Variable. (AM-15), Volume 15, will be forthcoming.

Functions of complex variables. --- Topology. --- 0O. --- 0Q. --- Absolute value. --- Addition. --- Additive group. --- Admissible representation. --- Arc (geometry). --- Arc length. --- Bernhard Riemann. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Branch point. --- Canonical form. --- Cartesian coordinate system. --- Central angle. --- Clockwise. --- Compact space. --- Concentric. --- Conformal map. --- Continuous function. --- Contour line. --- Convex set. --- Corollary. --- Countable set. --- Critical value. --- Curve. --- Deformation theory. --- Diameter. --- Euclidean vector. --- Euler characteristic. --- Existential quantification. --- Family of curves. --- Finitary. --- Finite set. --- Function (mathematics). --- Harmonic function. --- Homeomorphism. --- Homotopy. --- Inference. --- Integer. --- Interior (topology). --- Intersection (set theory). --- Isolated point. --- Jordan curve theorem. --- Length. --- Limit point. --- Maxima and minima. --- Maximal arc. --- Meromorphic function. --- Metric space. --- Minimum distance. --- Normal family. --- Parameter. --- Parametrization. --- Partial derivative. --- Picard theorem. --- Point at infinity. --- Polar coordinate system. --- Polynomial. --- Quantity. --- Rado's theorem (Ramsey theory). --- Real variable. --- Rectangle. --- Remainder. --- Riemann surface. --- Rigid body. --- Saddle point. --- Sequence. --- Set (mathematics). --- Similarity (geometry). --- Simply connected space. --- Special case. --- Subset. --- Summation. --- Theorem. --- Theory. --- Topological property. --- Unit circle. --- Unit vector. --- Without loss of generality.

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This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 10, 1970). Its unifying theme is the study of local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions that most closely resemble general complex analytic mappings in one complex dimension. The purpose of this volume is rather to clarify some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake.Originally published in 1970.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Complex analysis --- Analytic spaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Algebra homomorphism. --- Algebraic curve. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Associated prime. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Branch point. --- Change of variables. --- Characterization (mathematics). --- Codimension. --- Coefficient. --- Cohomology. --- Complete intersection. --- Complex analysis. --- Complex conjugate. --- Complex dimension. --- Complex number. --- Connected component (graph theory). --- Corollary. --- Critical point (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Disjoint union. --- Divisor. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Finitely generated module. --- Geometry. --- Hamiltonian mechanics. --- Holomorphic function. --- Homeomorphism. --- Homological dimension. --- Homomorphism. --- Hypersurface. --- Ideal (ring theory). --- Identity element. --- Induced homomorphism. --- Inequality (mathematics). --- Injective function. --- Integral domain. --- Invertible matrix. --- Irreducible component. --- Isolated singularity. --- Isomorphism class. --- Jacobian matrix and determinant. --- Linear map. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mathematics. --- Maximal element. --- Maximal ideal. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Module homomorphism. --- Monic polynomial. --- Monomial. --- Neighbourhood (mathematics). --- Noetherian. --- Open set. --- Parametric equation. --- Parametrization. --- Permutation. --- Polynomial ring. --- Polynomial. --- Power series. --- Quadratic form. --- Quotient module. --- Regular local ring. --- Removable singularity. --- Ring (mathematics). --- Ring homomorphism. --- Row and column vectors. --- Scalar multiplication. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Submanifold. --- Subset. --- Summation. --- Surjective function. --- Taylor series. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Vector space. --- Weierstrass preparation theorem. --- Zero divisor. --- Fonctions de plusieurs variables complexes --- Variétés complexes

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The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of construction, topological equivalence, and conformal mappings of one Riemann surface on another. The analytic part is concerned with the existence and properties of functions that have a special character connected with the conformal structure, for instance: subharmonic, harmonic, and analytic functions.Originally published in 1960.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

515.16 --- 515.16 Topology of manifolds --- Topology of manifolds --- Riemann surfaces. --- Topology. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Surfaces, Riemann --- Functions --- Analytic function. --- Axiom of choice. --- Basis (linear algebra). --- Betti number. --- Big O notation. --- Bijection. --- Bilinear form. --- Bolzano–Weierstrass theorem. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Canonical basis. --- Cauchy sequence. --- Cauchy's integral formula. --- Characterization (mathematics). --- Coefficient. --- Commutator subgroup. --- Compact space. --- Compactification (mathematics). --- Conformal map. --- Connected space. --- Connectedness. --- Continuous function (set theory). --- Continuous function. --- Coset. --- Cross-cap. --- Dirichlet integral. --- Disjoint union. --- Elementary function. --- Elliptic surface. --- Exact differential. --- Existence theorem. --- Existential quantification. --- Extremal length. --- Family of sets. --- Finite intersection property. --- Finitely generated abelian group. --- Free group. --- Function (mathematics). --- Fundamental group. --- Green's function. --- Harmonic differential. --- Harmonic function. --- Harmonic measure. --- Heine–Borel theorem. --- Homeomorphism. --- Homology (mathematics). --- Ideal point. --- Infimum and supremum. --- Isolated point. --- Isolated singularity. --- Jordan curve theorem. --- Lebesgue integration. --- Limit point. --- Line segment. --- Linear independence. --- Linear map. --- Maximal set. --- Maximum principle. --- Meromorphic function. --- Metric space. --- Normal operator. --- Normal subgroup. --- Open set. --- Orientability. --- Orthogonal complement. --- Partition of unity. --- Point at infinity. --- Polyhedron. --- Positive harmonic function. --- Principal value. --- Projection (linear algebra). --- Projection (mathematics). --- Removable singularity. --- Riemann mapping theorem. --- Riemann surface. --- Semi-continuity. --- Sign (mathematics). --- Simplicial homology. --- Simply connected space. --- Singular homology. --- Skew-symmetric matrix. --- Special case. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Taylor series. --- Theorem. --- Topological space. --- Triangle inequality. --- Uniform continuity. --- Uniformization theorem. --- Unit disk. --- Upper and lower bounds. --- Upper half-plane. --- Weyl's lemma (Laplace equation). --- Zorn's lemma.

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This book is meant to give an account of recent developments in the theory of Plateau's problem for parametric minimal surfaces and surfaces of prescribed constant mean curvature ("H-surfaces") and its analytical framework. A comprehensive overview of the classical existence and regularity theory for disc-type minimal and H-surfaces is given and recent advances toward general structure theorems concerning the existence of multiple solutions are explored in full detail.The book focuses on the author's derivation of the Morse-inequalities and in particular the mountain-pass-lemma of Morse-Tompkins and Shiffman for minimal surfaces and the proof of the existence of large (unstable) H-surfaces (Rellich's conjecture) due to Brezis-Coron, Steffen, and the author. Many related results are covered as well. More than the geometric aspects of Plateau's problem (which have been exhaustively covered elsewhere), the author stresses the analytic side. The emphasis lies on the variational method.Originally published in 1989.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Calculus of variations. --- Global analysis (Mathematics). --- Minimal surfaces. --- Plateau's problem. --- Global analysis (Mathematics) --- MATHEMATICS / Geometry / Differential. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Minimal surface problem --- Plateau problem --- Problem of Plateau --- Minimal surfaces --- Surfaces, Minimal --- Banach space. --- Bernhard Riemann. --- Big O notation. --- Boundary value problem. --- Branch point. --- C0. --- Closed geodesic. --- Compact space. --- Complex analysis. --- Complex number. --- Conformal map. --- Conjecture. --- Contradiction. --- Convex curve. --- Convex set. --- Differentiable function. --- Direct method in the calculus of variations. --- Dirichlet integral. --- Dirichlet problem. --- Embedding. --- Estimation. --- Euler–Lagrange equation. --- Existential quantification. --- Geometric measure theory. --- Global analysis. --- Jordan curve theorem. --- Linear differential equation. --- Mathematical analysis. --- Mathematical problem. --- Mathematician. --- Maximum principle. --- Mean curvature. --- Metric space. --- Minimal surface. --- Modulus of continuity. --- Morse theory. --- Nonparametric statistics. --- Normal (geometry). --- Parallel projection. --- Parameter space. --- Parametrization. --- Partial differential equation. --- Quadratic growth. --- Quantity. --- Riemann mapping theorem. --- Second derivative. --- Sign (mathematics). --- Special case. --- Surface area. --- Tangent space. --- Theorem. --- Total curvature. --- Uniform convergence. --- Variational method (quantum mechanics). --- Variational principle. --- W0. --- Weak solution.

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The description for this book, Seminar On Minimal Submanifolds. (AM-103), Volume 103, will be forthcoming.

Minimal submanifolds. --- A priori estimate. --- Analytic function. --- Banach space. --- Boundary (topology). --- Boundary value problem. --- Bounded set (topological vector space). --- Branch point. --- Cauchy–Riemann equations. --- Center manifold. --- Closed geodesic. --- Codimension. --- Coefficient. --- Cohomology. --- Compactness theorem. --- Comparison theorem. --- Configuration space. --- Conformal geometry. --- Conformal group. --- Conformal map. --- Continuous function. --- Cross product. --- Curve. --- Degeneracy (mathematics). --- Diffeomorphism. --- Differential form. --- Dirac operator. --- Discrete group. --- Divergence theorem. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation. --- Existence theorem. --- Existential quantification. --- Exterior derivative. --- First variation. --- Free boundary problem. --- Fundamental group. --- Gauss map. --- Geodesic. --- Geometry. --- Group action. --- Hamiltonian mechanics. --- Harmonic function. --- Harmonic map. --- Hausdorff dimension. --- Hausdorff measure. --- Homotopy group. --- Homotopy. --- Hurewicz theorem. --- Hyperbolic 3-manifold. --- Hyperbolic manifold. --- Hyperbolic space. --- Hypersurface. --- Implicit function theorem. --- Infimum and supremum. --- Injective function. --- Inner automorphism. --- Isolated singularity. --- Isometry group. --- Isoperimetric problem. --- Klein bottle. --- Kleinian group. --- Limit set. --- Lipschitz continuity. --- Mapping class group. --- Maxima and minima. --- Maximum principle. --- Minimal surface of revolution. --- Minimal surface. --- Monotonic function. --- Möbius transformation. --- Norm (mathematics). --- Orthonormal basis. --- Parametric surface. --- Periodic function. --- Poincaré conjecture. --- Projection (linear algebra). --- Regularity theorem. --- Riemann surface. --- Riemannian manifold. --- Schwarz reflection principle. --- Second fundamental form. --- Semi-continuity. --- Simply connected space. --- Special case. --- Stein's lemma. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subsequence. --- Support (mathematics). --- Symplectic manifold. --- Tangent space. --- Teichmüller space. --- Theorem. --- Trace (linear algebra). --- Uniformization. --- Uniqueness theorem. --- Variational principle. --- Yamabe problem.