2 edition of **Harmonic maps of spheres and equivariant theory** found in the catalog.

Harmonic maps of spheres and equivariant theory

Andrea Ratto

- 268 Want to read
- 26 Currently reading

Published
**1987**
by typescript in [s.l.]
.

Written in English

**Edition Notes**

Thesis (Ph.D.) - University of Warwick, 1987.

Statement | by Andrea Ratto. |

ID Numbers | |
---|---|

Open Library | OL13922192M |

Finally, the rudiments of an equivariant theory of harmonic maps having been set out earlier, we find that our examples can also be put in this framework.\ud The second significant result which arose from this study is a strong candidate for a counterexample: suppose Sn is stretched to a length b in one direction to make an ellipsoid En(b). ON NONRIGIDITY OF HARMONIC MAPS INTO SPHERES GABOR TOTH Abstract. This note studies nonrigidity of equivariant harmonic maps /: M -» S" of a Riemannian homogeneous space M into the Euclidean «-sphere S" via represen-tation theory applied to .

The first of three parts comprising Vol the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July (ISBN for the set is ). Part 1 begins with a problem list by S.T. Yau, successor to his list (Sem. HARMONIC MAPS: GENERAL THEORY, MAPS OF SURFACES, AND RELATED VARIATIONAL PROBLEMS Harmonic Maps and Morphisms from Spheres and Deformed Spheres,Y.-X. Dong S1-Valued Harmonic Maps with High Topological Degree, E. Sandier and M. Soret Harmonic Maps to Non-Locally Compact Spaces, R. Shoen.

1 Finite Mobius Groups.- Platonic Solids and Finite Rotation Groups.- Rotations and Moebius Transformations.- Invariant Forms.- Minimal Immersions of the 3-sphere into Spheres.- Minimal Imbeddings of Spherical Space Forms into Spheres.- Additional Topic: Klein's Theory of the Icosahedron.- 2 Moduli for Eigenmaps.- Spherical Harmonics.- Generalities on Eigenmaps. This theory has rich inteconnections with a variety of mathematical disciplines such as invariant theory, convex geometry, harmonic maps, and orthogonal multiplications. In this book, the author traces the development of the study of spherical minimal immersions over the past 30 plus years, including Takahashi's proof regarding the Reviews: 1.

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Cite this paper as: Uhlenbeck K.K. () Equivariant harmonic maps into spheres. In: Knill R.J., Kalka M., Sealey H.C.J. (eds) Harmonic by: The existence proof involves equivariant regularity theory. As an application we have several new examples of nontrivial harmonic maps between spheres, including harmonic maps representing all.

By minimizing in Sobolev spaces of mappings which are equivariant with respect to certain torus actions, we construct homotopically nontrivial harmonic maps between spheres. Doing so, we can represent the nontrivial elements of π n+1 (S n) (n⩾3) and of π n+2 (S n) (n⩾5 odd) by harmonic maps, as well as infinitely many elements of π n (S Cited by: 8.

In Chapter I we produce many new harmonic maps of spheres by the qualitative study of the pendulum equations for the join and the Hopf construction.

In particular, we obtain Corollary Let Φ1: Sp -> Sr be any harmonic homogeneous polynomial of degree greater or equal than two, and let Φ2 be the identity map id: Sq -> Sq. Then the (q+1)-suspension of Φ1 is harmonically representable. The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.

Examples of Harmonic Maps from Disks to Hemispheres () Variational Theory in Fibre Bundles: Examples () Constructions Twistorielles des Applications Harmoniques () Removable Singularities of Harmonic Maps () On Equivariant Harmonic Maps () Regularity of Certain Harmonic Maps () Gauss Maps of Surfaces () Minimal.

In the limit e → ∞ we regain the energy functional (with the unit of energy Rf 2 π /2) for harmonic maps between three-spheres which are equivariant with respect to the action of SO(3) group. In the first section of this paper we study the Dirichlet problem for equivariant (rotationally symmetric) p-harmonic maps from the Euclidean ball B m to the closed upper ellipsoid E m + (b) (p ≥ 2, m ≥ 3): in particular, we establish a condition which is necessary and sufficient for the existence of an equivariant smooth solution with prescribed boundary values.

In the representation theory of finite groups, a vector space equipped with a group that acts by linear transformations of the space is called a linear representation of the group. A linear map that commutes with the action is called an is, an intertwiner is just an equivariant linear map between two representations.

Alternatively, an intertwiner for representations of a group. According to our current on-line database, Andrea Ratto has 1 student and 2 descendants. We welcome any additional information. If you have additional information or corrections regarding this mathematician, please use the update submit students of this mathematician, please use the new data form, noting this mathematician's MGP ID of for the advisor ID.

Harmonic maps of spheres and equivariant theory. By Andrea Ratto. Abstract. In Chapter I we produce many new harmonic maps of spheres by the qualitative study of the pendulum equations for the join and the Hopf construction. In particular, we obtain\ud \ud Corollary \ud \ud Let Φ1: Sp -> Sr be any harmonic homogeneous polynomial of.

Get this from a library. Harmonic maps with symmetry, harmonic morphisms, and deformations of metrics. [P Baird] -- "The aim of this book is to construct harmonic maps between Riemannian manifolds, and in particular between spheres.

These maps have a delightful geometry associated with them - they preserve. But, for maps into positively curved manifolds, especially for harmonic maps between spheres, the PDE method is not successful.

By heat flow method the solution blows up at finite time (cf. [C-G], [D2], [C-D]). In this case any harmonic map is unstable (see [X1], [Le1]) and the direct method is not applicable.

Harmonic maps in Kähler geometry and deformation theory. Pages Kalka, M. Preview. Harmonic foliations. Pages Kamber, Franz W.

(et al.) Equivariant harmonic maps into spheres. Pages Uhlenbeck, Karen K. Preview. Book Title Harmonic Maps Book Subtitle. Smoothness of certain equivariant harmonic maps 7. THE GENERAL THEORY OF HARMONIC MORPHISMS General theory Examples and non-examples of harmonic morphisms Maps ~: (M,g)__.(N,h) where ~h has two distinct non-zero eigenvalues 8.

HARMONIC MORPHISMS DEFINED BY HOMOGENEOUS POLYNOMIALS. For basic notions and results of primitive, equivariant and (non-)isotropic harmonic maps we refer to the readers [2,3,6]. r n 2 Equivariant primitive harmonic maps in F (CP) Equivariant primitive harmonic maps Amap ψ from a Riemann surface M into a k-symmetric space G/K is called primitive if 1,0 j dψ(T M) ⊂ g, where we denote the.

"Spherical soap bubbles", isometric minimal immersions of round spheres into round spheres, or spherical immersions for short, belong to a fast growing and fascinating area between algebra and geometry. This theory has rich interconnections with a variety of mathematical disciplines such as.

Examples of harmonic maps from disks to hemispheres / with L. Lemaire Variational theory in fibre bundles: examples Constructions twistorielles des applications harmoniques / with S. Salamon Removable singularities of harmonic maps / with J.C. Polking On equivariant harmonic maps.

In Chapter I we produce many new harmonic maps of spheres by the qualitative study of the pendulum equations for the join and the Hopf construction. In particular, we obtain Corollary Let Φ1: Sp -> Sr be any harmonic homogeneous polynomial of degree greater or equal than two, and let Φ2 be the identity map id: Sq -> Sq.

Mathematical definition. Here the notion of the laplacian of a map is considered from three different perspectives. A map is called harmonic if its laplacian vanishes; it is called totally geodesic if its hessian vanishes. Integral formulation. Let (M, g) and (N, h) be Riemannian manifolds.

Given a smooth map f from M to N, the pullback f * h is a symmetric 2-tensor on M; the energy density e. This paper is about harmonic maps from closed Riemann surfaces into homogeneous spaces such as flag manifolds and loop groups.

It contains the construction of a family of new examples of harmonic maps from T 2 =S 1 ×S 1 into F(n) or Ω(U(n)) that are not holomorphic with respect to any almost complex structure on F(n) or Ω(U(n)), where F(n) is the quotient of U(n) by any maximal torus and Ω.

compact images, which contain wide class of non-equivariant harmonic maps. Similar arguments as proof of Theorem also yield stability of small size harmonic maps into Ka¨hler manifolds: Theorem Let M = H2, and N be a compact 2n-dimensional Kahler manifold. Let Q: H2 → N be an admissible harmonic map with compact image.

Given any δ.Harmonic maps in Kähler geometry and deformation theory.- Harmonic foliations.- On the stability of harmonic maps.- Stability of harmonic maps between symmetric spaces.- On a class of harmonic maps.- Harmonic diffeomorphisms of surfaces.- Equivariant harmonic maps into spheres.

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