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Part explanation of important recent work, and part introduction to some of the techniques of modern partial differential equations, this monograph is a self-contained exposition of the Neumann problem for the Cauchy-Riemann complex and certain of its applications. The authors prove the main existence and regularity theorems in detail, assuming only a knowledge of the basic theory of differentiable manifolds and operators on Hilbert space. They discuss applications to the theory of several complex variables, examine the associated complex on the boundary, and outline other techniques relevant to these problems. In an appendix they develop the functional analysis of differential operators in terms of Sobolev spaces, to the extent it is required for the monograph.

Functional analysis --- Neumann problem --- Differential operators --- Complex manifolds --- Complex manifolds. --- Differential operators. --- Neumann problem. --- Differential equations, Partial --- Équations aux dérivées partielles --- Analytic spaces --- Manifolds (Mathematics) --- Operators, Differential --- Differential equations --- Operator theory --- Boundary value problems --- A priori estimate. --- Almost complex manifold. --- Analytic function. --- Apply. --- Approximation. --- Bernhard Riemann. --- Boundary value problem. --- Calculation. --- Cauchy–Riemann equations. --- Cohomology. --- Compact space. --- Complex analysis. --- Complex manifold. --- Coordinate system. --- Corollary. --- Derivative. --- Differentiable manifold. --- Differential equation. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Estimation. --- Euclidean space. --- Existence theorem. --- Exterior (topology). --- Finite difference. --- Fourier analysis. --- Fourier transform. --- Frobenius theorem (differential topology). --- Functional analysis. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Irreducible representation. --- Line segment. --- Linear programming. --- Local coordinates. --- Lp space. --- Manifold. --- Monograph. --- Multi-index notation. --- Nonlinear system. --- Operator (physics). --- Overdetermined system. --- Partial differential equation. --- Partition of unity. --- Potential theory. --- Power series. --- Pseudo-differential operator. --- Pseudoconvexity. --- Pseudogroup. --- Pullback. --- Regularity theorem. --- Remainder. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Smoothness. --- Sobolev space. --- Special case. --- Statistical significance. --- Sturm–Liouville theory. --- Submanifold. --- Tangent bundle. --- Theorem. --- Uniform norm. --- Vector field. --- Weight function. --- Operators in hilbert space

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This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

Submanifolds. --- Functions of several complex variables. --- Algebraic equation. --- Algebraic function. --- Algebraic manifold. --- Algebraic variety. --- Analytic function. --- Analytic geometry. --- Antiholomorphic function. --- Arbitrarily large. --- Automorphism. --- Banach space. --- Biholomorphism. --- Boundary value problem. --- CR manifold. --- Calculation. --- Canonical coordinates. --- Cauchy sequence. --- Cauchy–Riemann equations. --- Change of variables. --- Codimension. --- Commutative algebra. --- Commutator. --- Complex analysis. --- Complex dimension. --- Complex number. --- Complex plane. --- Complex space. --- Complexification (Lie group). --- Complexification. --- Connected space. --- Continuous function. --- Counterexample. --- Degenerate bilinear form. --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Direct proof. --- Equation. --- Existential quantification. --- Exponential map (Lie theory). --- Field of fractions. --- First-order partial differential equation. --- Formal power series. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Function (mathematics). --- Geometry. --- Hermitian adjoint. --- Hilbert transform. --- Holomorphic function. --- Homogeneous coordinates. --- Hopf lemma. --- Hyperfunction. --- Hyperplane. --- Hypersurface. --- Implicit function theorem. --- Integrable system. --- Integral curve. --- Integral domain. --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Irreducible polynomial. --- Kobayashi metric. --- Lie algebra. --- Linear algebra. --- Linear subspace. --- Local diffeomorphism. --- Monodromy theorem. --- Neighbourhood (mathematics). --- Open set. --- Parametrization. --- Partial differential equation. --- Poisson kernel. --- Polynomial. --- Power series. --- Pseudoconvexity. --- Right inverse. --- Several complex variables. --- Special case. --- Stokes' theorem. --- Subbundle. --- Subharmonic function. --- Submanifold. --- Summation. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Taylor series. --- Theorem. --- Topological space. --- Topology. --- Transcendence degree. --- Transversal (geometry). --- Union (set theory). --- Unit vector. --- Variable (mathematics). --- Vector field. --- Vector space. --- Weierstrass preparation theorem.