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The book presents a collection of results pertaining to the partial regularity of solutions to various variational problems, all of which are connected to the Dirichlet energy of maps between Riemannian manifolds, and thus related to the harmonic map problem. The topics covered include harmonic maps and generalized harmonic maps; certain perturbed versions of the harmonic map equation; the harmonic map heat flow; and the Landau-Lifshitz (or Landau-Lifshitz-Gilbert) equation. Since the methods in regularity theory of harmonic maps are quite subtle, it is not immediately clear how they can be ap

Harmonic maps --- Mathematical physics. --- Physical mathematics --- Physics --- Maps, Harmonic --- Mappings (Mathematics) --- Mathematical models. --- Mathematics --- Mathematical physics --- Harmonic maps - Mathematical models.

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Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

Harmonic maps. --- Mappings (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- Maps, Harmonic

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Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Harmonic maps. --- Loops (Group theory) --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Loop groups --- Group theory --- Maps, Harmonic --- Mappings (Mathematics)

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"The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.The book begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichm©oller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification"--Provided by publisher.

Mappings (Mathematics) --- Class groups (Mathematics) --- Groups, Class (Mathematics) --- Algebraic number theory --- Commutative rings --- Ideals (Algebra) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- 3-manifold theory. --- Alexander method. --- Birman exact sequence. --- BirmanЈilden theorem. --- Dehn twists. --- DehnЌickorish theorem. --- DehnЎielsenЂaer theorem. --- Dennis Johnson. --- Euler class. --- FenchelЎielsen coordinates. --- Gervais presentation. --- Grtzsch's problem. --- Johnson homomorphism. --- Markov partitions. --- Meyer signature cocycle. --- Mod(S). --- Nielsen realization theorem. --- NielsenДhurston classification theorem. --- NielsenДhurston classification. --- Riemann surface. --- Teichmller mapping. --- Teichmller metric. --- Teichmller space. --- Thurston compactification. --- Torelli group. --- Wajnryb presentation. --- algebraic integers. --- algebraic intersection number. --- algebraic relations. --- algebraic structure. --- annulus. --- aspherical manifold. --- bigon criterion. --- braid group. --- branched cover. --- capping homomorphism. --- classifying space. --- closed surface. --- collar lemma. --- compactness criterion. --- complex of curves. --- configuration space. --- conjugacy class. --- coordinates principle. --- cutting homomorphism. --- cyclic subgroup. --- diffeomorphism. --- disk. --- existence theorem. --- extended mapping class group. --- finite index. --- finite subgroup. --- finite-order homeomorphism. --- finite-order mapping class. --- first homology group. --- geodesic laminations. --- geometric classification. --- geometric group theory. --- geometric intersection number. --- geometric operation. --- geometry. --- harmonic maps. --- holomorphic quadratic differential. --- homeomorphism. --- homological criterion. --- homotopy. --- hyperbolic geometry. --- hyperbolic plane. --- hyperbolic structure. --- hyperbolic surface. --- inclusion homomorphism. --- infinity. --- intersection number. --- isotopy. --- lantern relation. --- low-dimensional homology. --- mapping class group. --- mapping torus. --- measured foliation space. --- measured foliations. --- metric geometry. --- moduli space. --- orbifold. --- orbit. --- outer automorphism group. --- pseudo-Anosov homeomorphism. --- punctured disk. --- quasi-isometry. --- quasiconformal map. --- second homology group. --- simple closed curve. --- simplicial complex. --- stretch factors. --- surface bundles. --- surface homeomorphism. --- surface. --- symplectic representation. --- topology. --- torsion. --- torus. --- train track.