Harmonic maps, conservation laws, and moving frames [E-Book] / Frédéric Hélein.
Hélein, Frédéric, (author)
Second edition.
Cambridge : Cambridge University Press, 2002
1 online resource (xxv, 264 pages)
englisch
9780511543036
9780521811606
Cambridge tracts in mathematics ; 150
Full Text
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505 0 0 |g 1  |t Geometric and analytic setting  |g 1 --  |g 1.1  |t The Laplacian on (M, g)  |g 2 --  |g 1.2  |t Harmonic maps between two Riemannian manifolds  |g 5 --  |g 1.3  |t Conservation laws for harmonic maps  |g 11 --  |g 1.3.1  |t Symmetries on N  |g 12 --  |g 1.3.2  |t Symmetries on M: the stress-energy tensor  |g 18 --  |g 1.3.3  |t Consequences of theorem 1.3.6  |g 24 --  |g 1.4  |t Variational approach: Sobolev spaces  |g 31 --  |g 1.4.1  |t Weakly harmonic maps  |g 37 --  |g 1.4.2  |t Weakly Noether harmonic maps  |g 42 --  |g 1.4.3  |t Minimizing maps  |g 42 --  |g 1.4.4  |t Weakly stationary maps  |g 43 --  |g 1.4.5  |t Relation between these different definitions  |g 43 --  |g 1.5  |t Regularity of weak solutions  |g 46 --  |g 2  |t Harmonic maps with symmetry  |g 49 --  |g 2.1  |t Backlund transformation  |g 50 --  |g 2.1.1  |t S[superscript 2]-valued maps  |g 50 --  |g 2.1.2  |t Maps taking values in a sphere S[superscript n], n [greater than or equal] 2  |g 54 --  |g 2.1.3  |t Comparison  |g 56 --  |g 2.2  |t Harmonic maps with values into Lie groups  |g 58 --  |g 2.2.1  |t Families of curvature-free connections  |g 65 --  |g 2.2.2  |t The dressing  |g 72 --  |g 2.2.3  |t Uhlenbeck factorization for maps with values in U(n)  |g 77 --  |g 2.2.4  |t S[superscript 1]-action  |g 79 --  |g 2.3  |t Harmonic maps with values into homogeneous spaces  |g 82 --  |g 2.4  |t Synthesis: relation between the different formulations  |g 95 --  |g 2.5  |t Compactness of weak solutions in the weak topology  |g 101 --  |g 2.6  |t Regularity of weak solutions  |g 109 --  |g 3  |t Compensations and exotic function spaces  |g 114 --  |g 3.1  |t Wente's inequality  |g 115 --  |g 3.1.1  |t The inequality on a plane domain  |g 115 --  |g 3.1.2  |t The inequality on a Riemann surface  |g 119 --  |g 3.2  |t Hardy spaces  |g 128 --  |g 3.3  |t Lorentz spaces  |g 135 --  |g 3.4  |t Back to Wente's inequality  |g 145 --  |g 3.5  |t Weakly stationary maps with values into a sphere  |g 150 --  |g 4  |t Harmonic maps without symmetry  |g 165 --  |g 4.1  |t Regularity of weakly harmonic maps of surfaces  |g 166 --  |g 4.2  |t Generalizations in dimension 2  |g 187 --  |g 4.3  |t Regularity results in arbitrary dimension  |g 193 --  |g 4.4  |t Conservation laws for harmonic maps without symmetry  |g 205 --  |g 4.4.1  |t Conservation laws  |g 206 --  |g 4.4.2  |t Isometric embedding of vector-bundle-valued differential forms  |g 211 --  |g 4.4.3  |t A variational formulation for the case m = n = 2 and p = 1  |g 215 --  |g 4.4.4  |t Hidden symmetries for harmonic maps on surfaces?  |g 218 --  |g 5  |t Surfaces with mean curvature in L[superscript 2]  |g 221 --  |g 5.1  |t Local results  |g 224 --  |g 5.2  |t Global results  |g 237 --  |g 5.3  |t Willmore surfaces  |g 242 --  |g 5.4  |t Epilogue: Coulomb frames and conformal coordinates  |g 244. 
520 |a The author presents an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A self-contained presentation of 'exotic' functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a 'Coulomb moving frame' is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces. 
650 0 |a Harmonic maps. 
650 0 |a Riemannian manifolds. 
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