scholarly journals ADDING A UNITON VIA THE DPW METHOD

2009 ◽  
Vol 20 (08) ◽  
pp. 997-1010 ◽  
Author(s):  
N. CORREIA ◽  
R. PACHECO
Keyword(s):  
Finite Type ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Point Of View ◽  
Type Property ◽  
Riemannian Symmetric Spaces

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

scholarly journals Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings

2011 ◽  
Vol 11 (2) ◽  
pp. 421-465
Author(s):  
Bertrand Rémy ◽  
Amaury Thuillier ◽  
Annette Werner
Keyword(s):  
Symmetric Spaces ◽  
Linear Representation ◽  
Linear Algebraic Group ◽  
Point Of View ◽  
Analytic Geometry ◽  
The Third ◽  
Equivariant Embedding ◽  
Tits Building ◽  
Riemannian Symmetric Spaces ◽  
Special Case

AbstractIn the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

scholarly journals Integrability and vesture for harmonic maps into symmetric spaces

2016 ◽  
Vol 28 (03) ◽  
pp. 1650006 ◽  
Author(s):  
Shabnam Beheshti ◽  
Shadi Tahvildar-Zadeh
Keyword(s):  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Dressing Method ◽  
Special Cases ◽  
Einstein Vacuum ◽  
Completely Reducible

After formulating the notion of integrability for axially symmetric harmonic maps from [Formula: see text] into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding [Formula: see text]-solitonic harmonic maps into a non-compact Grassmann manifold [Formula: see text] is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases [Formula: see text] and [Formula: see text] and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

BOCHNER-TYPE FORMULAS FOR TRANSVERSALLY HARMONIC MAPS

2012 ◽  
Vol 23 (03) ◽  
pp. 1250003 ◽  
Author(s):  
QUN CHEN ◽  
WUBIN ZHOU
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Transverse Curvature ◽  
Sasaki Manifold ◽  
Sasaki Manifolds

The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M′, and M′ has a strongly negative transverse curvature. If the rank of dTf is at least three at some points of M, then f is contact holomorphic (or contact anti-holomorphic).

scholarly journals Wave propagation on Riemannian symmetric spaces

1992 ◽  
Vol 107 (2) ◽  
pp. 270-278 ◽  
Author(s):  
G. 'Olafsson ◽  
H. Schlichtkrull
Keyword(s):  
Wave Propagation ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces
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