By Frédéric Hélein
This available creation to harmonic map concept and its analytical features, covers contemporary advancements within the regularity concept of weakly harmonic maps. The booklet starts by way of introducing those ideas, stressing the interaction among geometry, the function of symmetries and susceptible strategies. It then provides a guided journey into the idea of thoroughly integrable platforms for harmonic maps, via chapters dedicated to fresh effects at the regularity of vulnerable strategies. A presentation of "exotic" useful areas from the speculation of harmonic research is given and those instruments are then used for proving regularity effects. the significance of conservation legislation is under pressure and the concept that of a "Coulomb relocating body" is defined intimately. The booklet ends with additional purposes and illustrations of Coulomb relocating frames to the speculation of surfaces.
Read or Download Harmonic maps, conservation laws, and moving frames PDF
Similar topology books
Goals to inspire mathematicians to demonstrate their paintings and to assist artists comprehend the tips expressed through such drawings. This publication explains the photograph layout of illustrations from Thurston's global of low-dimensional geometry and topology. It provides the rules of linear and aerial point of view from the point of view of projective geometry.
This quantity includes contributions by way of 3 authors and treats elements of noncommutative geometry which are regarding cyclic homology. The authors supply quite entire debts of cyclic concept from diverse and complementary issues of view. The connections among topological (bivariant) K-theory and cyclic thought through generalized Chern-characters are mentioned intimately.
This quantity comprises the lawsuits of the Workshop on Topology held on the Pontif? cia Universidade Cat? lica in Rio de Janeiro in January 1992. Bringing jointly approximately one hundred mathematicians from Brazil and all over the world, the workshop coated various themes in differential and algebraic topology, together with staff activities, foliations, low-dimensional topology, and connections to differential geometry.
This textbook on simple topology features a precise advent to normal topology and an creation to algebraic topology through its so much classical and trouble-free section based on the notions of basic workforce and protecting area. The e-book is customized for the reader who's made up our minds to paintings actively.
- Ordering Braids (Mathematical Surveys and Monographs)
- Topology (University mathematical texts)
- Topics In Topology And Homotopy Theory
- Fibre Bundles
Additional info for Harmonic maps, conservation laws, and moving frames
2. 2 The Dirichlet integral, and the set of harmonic maps on an open subset of a Riemannian surface, depend only on the conformal structure. 3, we can always suppose that we have locally conformal coordinates (x, y) ∈ R2 on (M, g). 13) becomes ∂ 2 ui ∂ 2 ui + + Γijk (u) 2 ∂x ∂y 2 ∂uj ∂uk ∂uj ∂uk + ∂x ∂x ∂y ∂y = 0. 13), but an alternative one where we think of N as a submanifold of a Euclidean space. In fact, thanks to the Nash–Moser theorem (, , ), we know that, provided h is C 3 , it is always possible to isometrically embed (N , h) into a vector space RN , with the Euclidean scalar product .
In the case where m = 2, this method no longer works, but Luc Lemaire proved a similar result which is valid for every simply connected open set . 44) which we used in Case A to deduce a conservation law in the case where there exists a Killing vector field. Writing Sβα = g αγ Sγβ , this identity states that, for any vector field X, 2Sαβ ∇ ∂ ∂xβ X α = −(LX g αβ )Sαβ . 37) implies that ∇ ∂α Sβα = 0 for ∂x all β, and thus ∇ ∂ ∂xβ (Sαβ X α ) = Sαβ ∇ ∂ ∂xβ Xα + ∇ 1 = − (LX g αβ )Sαβ . 52) Geometric and analytic setting 28 We see that for the vector field Y = Sαβ X α ∂x∂ β to be covariant divergencefree, it is necessary and sufficient that (LX g αβ )Sαβ = 0 .
1 Let X be a Lipschitz tangent vector field on N , which is an infinitesimal symmetry for L. 29) or equivalently, using the coordinates (x1 , . . , xm ) on Ω such that dµ = ρ(x)dx1 . . dxm , m ∂ α ∂x α=1 ρ(x)X i (u) ∂L (x, u, du) ∂Aiα = 0. 2 (i) In the case where L is the Lagrangian of the harmonic map, this result was first obtained in . (ii) The vector field J defined over Ω by J α = ρ(x) ∂L (x, u, du)X i (u) ∂Aiα is often called the Noether current by physicists. 30) are called conservation laws.
Harmonic maps, conservation laws, and moving frames by Frédéric Hélein