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
Table of Contents:
  • 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.