By Frédéric Hélein

ISBN-10: 0511016603

ISBN-13: 9780511016608

ISBN-10: 0521811600

ISBN-13: 9780521811606

This obtainable creation to harmonic map thought and its analytical points, covers fresh advancements within the regularity concept of weakly harmonic maps. The ebook starts by way of introducing those ideas, stressing the interaction among geometry, the function of symmetries and vulnerable suggestions. It then provides a guided travel into the speculation of thoroughly integrable platforms for harmonic maps, by means of chapters dedicated to fresh effects at the regularity of susceptible 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 idea that of a "Coulomb relocating body" is defined intimately. The publication ends with extra purposes and illustrations of Coulomb relocating frames to the speculation of surfaces.

**Read Online or Download Harmonic maps, conservation laws, and moving frames PDF**

**Best topology books**

**Download PDF by Frédéric Hélein: Harmonic maps, conservation laws, and moving frames**

This available creation to harmonic map idea and its analytical points, covers contemporary advancements within the regularity idea of weakly harmonic maps. The publication starts off via introducing those suggestions, stressing the interaction among geometry, the function of symmetries and susceptible suggestions. It then offers a guided journey into the idea of thoroughly integrable structures for harmonic maps, by way of chapters dedicated to contemporary effects at the regularity of susceptible recommendations.

**Paul A. Smith, Samuel Eilenberg's Point set topology PDF**

Appropriate for an entire path in topology, this article additionally capabilities as a self-contained therapy for self reliant learn. extra enrichment fabrics and complex subject coverage—including huge fabric on differentiable manifolds, summary harmonic research, and glued element theorems—constitute an exceptional reference for arithmetic lecturers, scholars, and execs.

**Download e-book for iPad: A homology theory for Smale spaces by Ian F. Putnam**

The writer develops a homology conception for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it's in accordance with parts. the 1st is a higher model of Bowen's outcome that each such method is identical to a shift of finite style less than a finite-to-one issue map. the second one is Krieger's size workforce invariant for shifts of finite variety.

- Classical sequences in Banach spaces
- Differential Topology and Geometry: Proceedings of the Colloquium Held at Dijon, 17–22 June, 1974
- Knot Theory (Mathematical Association of America Textbooks)
- Categories and Grothendieck Topologies [Lecture notes]

**Extra info for Harmonic maps, conservation laws, and moving frames**

**Sample text**

The first is obtained in the case where the symmetry group acts on the image manifold, and the second is connected to the symmetries of the domain. 1 Symmetries on N We start with a simple example, that of harmonic maps from an open set Ω of Rm , taking values in the sphere S 2 ⊂ R3 . 24) where |du|2 = 3 m i=1 α=1 ∂ui ∂xα 2 = 2e(u), and m ∆u = ∂2u . 25) or equivalently, m ∂ α ∂x α=1 u× ∂u ∂xα m = ∂u ∂u × α + u × ∆u = 0 . 26). Its interest is that it singles out a vector field j = (j 1 , . . , j m ) on Ω, given by jα = u × ∂u ∂xα which is divergence-free.

54) The interest of this theorem is that the group of conformal transformations of a Riemannian surface is very big. 3 Conservation laws for harmonic maps 29 and every holomorphic (or anti-holomorphic) diffeomorphism over ω induces a conformal transformation. Thus, for every holomorphic function F : ω −→ C, the vector field X = (F ) ∂ ∂ + (F ) ∂x ∂y is a conformal Killing field. In these coordinates (Sαβ ) = |du|2 2 = 1 2 1 0 0 1 |ux |2 u x , uy − − φ φ φ φ |du|2 = ∂u ∂x u x , uy |uy |2 , where 2 + ∂u ∂y 2 , and φ= ∂u ∂x 2 − ∂u ∂y 2 − 2i ∂u ∂u , ∂x ∂y .

Still in the case m = 2, we have ∆e2v g (φ) = e−2v ∆g φ . 9) Therefore, for m = 2, every function which is harmonic over (M, g) will also be so over (M, e2v g). 11) and where λ= ∂T i ∂T j 1 αβ g (x)gij (T (x)) α β . 2 The Dirichlet integral, and the set of harmonic functions over an open subset of a Riemannian surface, depend only on the conformal structure of this surface. This phenomenon, characteristic of dimension 2, has many consequences, among them the following, which is very useful: first recall that according to the theorem below, locally all conformal structures are equivalent.

### Harmonic maps, conservation laws, and moving frames by Frédéric Hélein

by Ronald

4.3