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Mathematical analysis --- Analyse mathématique --- Cauchy transform. --- Transformation de Cauchy. --- Cauchy-Riemann equations. --- Cauchy-Riemann, Équations de. --- Cauchy problem. --- Cauchy, Problème de. --- Mathematical physics --- Physique mathématique --- Analyse mathématique --- Cauchy-Riemann, Équations de. --- Cauchy, Problème de. --- Physique mathématique

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This book covers the necessary topics for learning the basic properties of complex manifolds and their submanifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications. The book begins with an introduction to the geometry of complex manifolds and their submanifolds and describes the properties of hypersurfaces and CR submanifolds, with particular emphasis on CR submanifolds of maximal CR dimension. The second part contains results which are not new, but recently published in some mathematical journals. The final part contains several original results by the authors, with complete proofs. Key features of "CR Submanifolds of Complex Projective Space": - Presents recent developments and results in the study of submanifolds previously published only in research papers. - Special topics explored include: the Kähler manifold, submersion and immersion, codimension reduction of a submanifold, tubes over submanifolds, geometry of hypersurfaces and CR submanifolds of maximal CR dimension. - Provides relevant techniques, results and their applications, and presents insight into the motivations and ideas behind the theory. - Presents the fundamental definitions and results necessary for reaching the frontiers of research in this field. This text is largely self-contained. Prerequisites include basic knowledge of introductory manifold theory and of curvature properties of Riemannian geometry. Advanced undergraduates, graduate students and researchers in differential geometry will benefit from this concise approach to an important topic.

Mathematics. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Several Complex Variables and Analytic Spaces. --- Global analysis. --- Differential equations, partial. --- Global differential geometry. --- Mathématiques --- Géométrie différentielle globale --- CR submanifolds. --- Cauchy-Riemann submanifolds --- Submanifolds, CR --- Manifolds (Mathematics) --- Cauchy-Riemann equations. --- Functions of several complex variables. --- CR submanifolds --- Mathematics --- Geometry --- Physical Sciences & Mathematics --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of complex variables. --- Differential geometry. --- Topology. --- Geometry, Differential --- Topology --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Partial differential equations --- Differential geometry --- Complex variables --- Elliptic functions --- Functions of real variables --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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The ∂̄ Neumann problem is probably the most important and natural example of a non-elliptic boundary value problem, arising as it does from the Cauchy-Riemann equations. It has been known for some time how to prove solvability and regularity by the use of L2 methods. In this monograph the authors apply recent methods involving the Heisenberg group to obtain parametricies and to give sharp estimates in various function spaces, leading to a better understanding of the ∂̄ Neumann problem. The authors have added substantial background material to make the monograph more accessible to students.Originally published in 1977.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.

Partial differential equations --- Neumann problem. --- Neumann problem --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Boundary value problems --- Differential equations, Partial --- A priori estimate. --- Abuse of notation. --- Analytic continuation. --- Analytic function. --- Approximation. --- Asymptotic expansion. --- Asymptotic formula. --- Basis (linear algebra). --- Besov space. --- Boundary (topology). --- Boundary value problem. --- Boundedness. --- Calculation. --- Cauchy's integral formula. --- Cauchy–Riemann equations. --- Change of variables. --- Characterization (mathematics). --- Combination. --- Commutative property. --- Commutator. --- Complex analysis. --- Complex manifold. --- Complex number. --- Computation. --- Convolution. --- Coordinate system. --- Corollary. --- Counterexample. --- Derivative. --- Determinant. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Dimensional analysis. --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic boundary value problem. --- Equation. --- Error term. --- Estimation. --- Even and odd functions. --- Existential quantification. --- Function space. --- Fundamental solution. --- Green's theorem. --- Half-space (geometry). --- Hardy's inequality. --- Heisenberg group. --- Holomorphic function. --- Infimum and supremum. --- Integer. --- Integral curve. --- Integral expression. --- Inverse function. --- Invertible matrix. --- Iteration. --- Laplace's equation. --- Left inverse. --- Lie algebra. --- Lie group. --- Linear combination. --- Logarithm. --- Lp space. --- Mathematical induction. --- Neumann boundary condition. --- Notation. --- Open problem. --- Orthogonal complement. --- Orthogonality. --- Parametrix. --- Partial derivative. --- Pointwise. --- Polynomial. --- Principal branch. --- Principal part. --- Projection (linear algebra). --- Pseudo-differential operator. --- Quantity. --- Recursive definition. --- Schwartz space. --- Scientific notation. --- Second derivative. --- Self-adjoint. --- Singular value. --- Sobolev space. --- Special case. --- Standard basis. --- Stein manifold. --- Subgroup. --- Subset. --- Summation. --- Support (mathematics). --- Tangent bundle. --- Theorem. --- Theory. --- Upper half-plane. --- Variable (mathematics). --- Vector field. --- Volume element. --- Weak solution. --- Neumann, Problème de --- Equations aux derivees partielles --- Problemes aux limites

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The description for this book, Recent Developments in Several Complex Variables. (AM-100), Volume 100, will be forthcoming.

Complex analysis --- Functions of several complex variables. --- Complex variables --- Several complex variables, Functions of --- Functions of complex variables --- Analytic continuation. --- Analytic function. --- Analytic set. --- Analytic space. --- Asymptotic expansion. --- Automorphic function. --- Axiom. --- Base change. --- Bergman metric. --- Betti number. --- Big O notation. --- Bilinear form. --- Boundary value problem. --- CR manifold. --- Canonical bundle. --- Cauchy problem. --- Cauchy–Riemann equations. --- Characteristic variety. --- Codimension. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Commutative property. --- Commutator. --- Compactification (mathematics). --- Complete intersection. --- Complete metric space. --- Complex dimension. --- Complex manifold. --- Complex number. --- Complex plane. --- Complex projective space. --- Complex space. --- Complex-analytic variety. --- Degeneracy (mathematics). --- Dense set. --- Determinant. --- Diffeomorphism. --- Differentiable function. --- Dimension (vector space). --- Dimension. --- Eigenvalues and eigenvectors. --- Embedding. --- Existential quantification. --- Explicit formulae (L-function). --- Fermat curve. --- Fiber bundle. --- Fundamental solution. --- Gorenstein ring. --- Hartogs' extension theorem. --- Hilbert space. --- Hilbert transform. --- Holomorphic function. --- Homotopy. --- Hyperfunction. --- Hypersurface. --- Hypoelliptic operator. --- Interpolation theorem. --- Irreducible component. --- Isometry. --- Linear map. --- Manifold. --- Maximal ideal. --- Monic polynomial. --- Monotonic function. --- Multiple integral. --- Nilpotent Lie algebra. --- Norm (mathematics). --- Open set. --- Orthogonal group. --- Parametrization. --- Permutation. --- Plurisubharmonic function. --- Polynomial. --- Principal bundle. --- Principal part. --- Principal value. --- Projection (linear algebra). --- Projective line. --- Proper map. --- Quadratic function. --- Real projective space. --- Resolution of singularities. --- Riemann surface. --- Riemannian manifold. --- Sectional curvature. --- Sheaf cohomology. --- Special case. --- Submanifold. --- Subset. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Uniqueness theorem. --- Unit disk. --- Unit sphere. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Fonctions de variables complexes --- Colloque

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This work is a fresh presentation of the Ahlfors-Weyl theory of holomorphic curves that takes into account some recent developments in Nevanlinna theory and several complex variables. The treatment is differential geometric throughout, and assumes no previous acquaintance with the classical theory of Nevanlinna. The main emphasis is on holomorphic curves defined over Riemann surfaces, which admit a harmonic exhaustion, and the main theorems of the subject are proved for such surfaces. The author discusses several directions for further research.

Analytic functions. --- Functions, Meromorphic. --- Value distribution theory. --- Meromorphic functions --- Functions, Analytic --- Functions, Monogenic --- Functions, Regular --- Regular functions --- Functions of complex variables --- Series, Taylor's --- Distribution of values theory --- Functions, Entire --- Functions, Meromorphic --- Addition. --- Algebraic curve. --- Algebraic number. --- Atlas (topology). --- Binomial coefficient. --- Cauchy–Riemann equations. --- Compact Riemann surface. --- Compact space. --- Complex manifold. --- Complex projective space. --- Computation. --- Continuous function (set theory). --- Covariant derivative. --- Critical value. --- Curvature form. --- Diagram (category theory). --- Differential form. --- Differential geometry of surfaces. --- Differential geometry. --- Dimension. --- Divisor. --- Essential singularity. --- Euler characteristic. --- Existential quantification. --- Fiber bundle. --- Gaussian curvature. --- Geodesic curvature. --- Geometry. --- Grassmannian. --- Harmonic function. --- Hermann Weyl. --- Hermitian manifold. --- Holomorphic function. --- Homology (mathematics). --- Hyperbolic manifold. --- Hyperplane. --- Hypersurface. --- Improper integral. --- Intersection number (graph theory). --- Isometry. --- Line integral. --- Manifold. --- Meromorphic function. --- Minimal surface. --- Nevanlinna theory. --- One-form. --- Open problem. --- Open set. --- Orthogonal complement. --- Parameter. --- Picard theorem. --- Product metric. --- Q.E.D. --- Remainder. --- Riemann sphere. --- Riemann surface. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent space. --- Tangent. --- Theorem. --- Three-dimensional space (mathematics). --- Unit circle. --- Unit vector. --- Vector field. --- Volume element. --- Volume form. --- Fonctions de plusieurs variables complexes