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Harmonic maps, conservation laws, and moving frames [electronic resource] / Fr�ed�eric H�elein.

By: H�elein, Fr�ed�eric, 1963-.
Material type: TextTextSeries: Cambridge tracts in mathematics: 150.Publisher: Cambridge ; New York : Cambridge University Press, 2002Edition: 2nd ed.Description: 1 online resource (xxv, 264 p.).ISBN: 0511016603 (electronic bk.); 9780511016608 (electronic bk.); 9780511543036 (electronic bk.); 0511543034 (electronic bk.).Subject(s): Harmonic maps | Riemannian manifolds | MATHEMATICS -- Topology | Applications harmoniques | Riemann, Vari�et�es de | Erhaltungssatz | Harmonische AbbildungGenre/Form: Electronic books.Additional physical formats: Print version:: Harmonic maps, conservation laws, and moving frames.DDC classification: 514/.74 Other classification: SK 370 Online resources: EBSCOhost
Contents:
Geometric and analytic setting The Laplacian on (M, g) Harmonic maps between two Riemannian manifolds Conservation laws for harmonic maps Symmetries on N Symmetries on M: the stress-energy tensor Consequences of theorem 1.3.6 Variational approach: Sobolev spaces Weakly harmonic maps Weakly Noether harmonic maps Minimizing maps Weakly stationary maps Relation between these different definitions Regularity of weak solutions Harmonic maps with symmetry Backlund transformation S[superscript 2]-valued maps Maps taking values in a sphere S[superscript n], n [greater than or equal] 2 Comparison Harmonic maps with values into Lie groups Families of curvature-free connections The dressing Uhlenbeck factorization for maps with values in U(n) S[superscript 1]-action Harmonic maps with values into homogeneous spaces Synthesis: relation between the different formulations Compactness of weak solutions in the weak topology Regularity of weak solutions Compensations and exotic function spaces Wente's inequality The inequality on a plane domain The inequality on a Riemann surface Hardy spaces Lorentz spaces Back to Wente's inequality Weakly stationary maps with values into a sphere Harmonic maps without symmetry Regularity of weakly harmonic maps of surfaces Generalizations in dimension 2 Regularity results in arbitrary dimension Conservation laws for harmonic maps without symmetry Conservation laws Isometric embedding of vector-bundle-valued differential forms A variational formulation for the case m = n = 2 and p = 1 Hidden symmetries for harmonic maps on surfaces? Surfaces with mean curvature in L[superscript 2] Local results Global results Willmore surfaces Epilogue: Coulomb frames and conformal coordinates
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Includes bibliographical references (p. 254-262) and index.

Description based on print version record.

1 Geometric and analytic setting 1 -- 1.1 The Laplacian on (M, g) 2 -- 1.2 Harmonic maps between two Riemannian manifolds 5 -- 1.3 Conservation laws for harmonic maps 11 -- 1.3.1 Symmetries on N 12 -- 1.3.2 Symmetries on M: the stress-energy tensor 18 -- 1.3.3 Consequences of theorem 1.3.6 24 -- 1.4 Variational approach: Sobolev spaces 31 -- 1.4.1 Weakly harmonic maps 37 -- 1.4.2 Weakly Noether harmonic maps 42 -- 1.4.3 Minimizing maps 42 -- 1.4.4 Weakly stationary maps 43 -- 1.4.5 Relation between these different definitions 43 -- 1.5 Regularity of weak solutions 46 -- 2 Harmonic maps with symmetry 49 -- 2.1 Backlund transformation 50 -- 2.1.1 S[superscript 2]-valued maps 50 -- 2.1.2 Maps taking values in a sphere S[superscript n], n [greater than or equal] 2 54 -- 2.1.3 Comparison 56 -- 2.2 Harmonic maps with values into Lie groups 58 -- 2.2.1 Families of curvature-free connections 65 -- 2.2.2 The dressing 72 -- 2.2.3 Uhlenbeck factorization for maps with values in U(n) 77 -- 2.2.4 S[superscript 1]-action 79 -- 2.3 Harmonic maps with values into homogeneous spaces 82 -- 2.4 Synthesis: relation between the different formulations 95 -- 2.5 Compactness of weak solutions in the weak topology 101 -- 2.6 Regularity of weak solutions 109 -- 3 Compensations and exotic function spaces 114 -- 3.1 Wente's inequality 115 -- 3.1.1 The inequality on a plane domain 115 -- 3.1.2 The inequality on a Riemann surface 119 -- 3.2 Hardy spaces 128 -- 3.3 Lorentz spaces 135 -- 3.4 Back to Wente's inequality 145 -- 3.5 Weakly stationary maps with values into a sphere 150 -- 4 Harmonic maps without symmetry 165 -- 4.1 Regularity of weakly harmonic maps of surfaces 166 -- 4.2 Generalizations in dimension 2 187 -- 4.3 Regularity results in arbitrary dimension 193 -- 4.4 Conservation laws for harmonic maps without symmetry 205 -- 4.4.1 Conservation laws 206 -- 4.4.2 Isometric embedding of vector-bundle-valued differential forms 211 -- 4.4.3 A variational formulation for the case m = n = 2 and p = 1 215 -- 4.4.4 Hidden symmetries for harmonic maps on surfaces? 218 -- 5 Surfaces with mean curvature in L[superscript 2] 221 -- 5.1 Local results 224 -- 5.2 Global results 237 -- 5.3 Willmore surfaces 242 -- 5.4 Epilogue: Coulomb frames and conformal coordinates 244.

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Other editions of this work

Harmonic maps, conservation laws and moving frames by Helein Frederic
Harmonic maps, conservation laws, and moving frames by H�elein, Fr�ed�eric, ©2002
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