On the stability of harmonic maps

1982 ◽  
pp. 122-129 ◽  
Author(s):  
Pui-Fai Leung
Keyword(s):  
Harmonic Maps ◽  
The Stability

Stability and constant boundary-value problems of harmonic maps with potential

2000 ◽  
Vol 68 (2) ◽  
pp. 145-154 ◽  
Author(s):  
Qun Chen
Keyword(s):  
Riemannian Manifold ◽  
Energy Density ◽  
Boundary Value Problem ◽  
Riemannian Manifolds ◽  
General Class ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Boundary Value ◽  
The Stability ◽  
Harmonic Maps With Potential

AbstractLet M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

scholarly journals On the stability of harmonic maps under the homogeneous Ricci flow

2018 ◽  
Vol 5 (1) ◽  
pp. 122-132
Author(s):  
Rafaela F. do Prado ◽  
Lino Grama
Keyword(s):  
Ricci Flow ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
The Stability

Abstract In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow.We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.

scholarly journals Harmonic Maps and Stability onf-Kenmotsu Manifolds

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Vittorio Mangione
Keyword(s):  
Harmonic Maps ◽  
Riemannian Submersions ◽  
Kenmotsu Manifold ◽  
Holomorphic Map ◽  
Kenmotsu Manifolds ◽  
The Stability ◽  
Kählerian Manifold

The purpose of this paper is to study some submanifolds and Riemannian submersions on anf-Kenmotsu manifold. The stability of aϕ-holomorphic map from a compactf-Kenmotsu manifold to a Kählerian manifold is proven.

Exponentially harmonic maps, exponential stress energy and stability

2016 ◽  
Vol 18 (06) ◽  
pp. 1550076 ◽  
Author(s):  
Yuan-Jen Chiang
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Stress Energy ◽  
The Stability

We study the exponential stress energy associated to an exponentially harmonic map between Riemannian manifolds. We prove three equivalent statements for a horizontally weakly conformal exponentially harmonic map between Riemannian manifolds. We also investigate the stability of exponentially harmonic maps.

scholarly journals Stability and Behaviour of Rotationally Symmetric Harmonic Maps From A Ball Into A Sphere

d'CARTESIAN ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 42
Author(s):  
Chriestie Montolalu
Keyword(s):  
Differential Equation ◽  
Harmonic Maps ◽  
Rotational Symmetry ◽  
Pendulum Equation ◽  
Rotationally Symmetric ◽  
The Stability

Rotationally Symmetric Harmonic Maps from a Ball into a Sphere has been studied before. The systems conducted in this study can be analyzed further by checking its stability and its behavior in the system. This paper will show how to determine the stability of the system and its behaviour by reducing it into a damped pendulum equation differential equation. Keywords:  Rotational symmetry, Harmonic maps, Stability, Damped pendulum equation.

STABLE HARMONIC MAPS BETWEEN FINSLER MANIFOLDS AND SSU MANIFOLDS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250015 ◽  
Author(s):  
JINTANG LI
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
American Mathematical Society ◽  
Variational Formula ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Recent Developments ◽  
Riemannian Case ◽  
The Stability ◽  
The Second Variation

Using the properties of Cartan tensor, we rewrite the second variation formula for harmonic maps between Finsler manifolds, and we prove that there is no non-degenerate stable harmonic map from a compact SSU manifold to any Finsler manifold, which is obtained by Howard and Wei for the Riemannian case. We also include a proof of a theorem of Shen–Wei which states that there is no non-degenerate stable harmonic map from a compact Finsler manifold to any SSU manifold, by the same second variational formula (see Eq. (2.1) in [Y. B. Shen and S. W. Wei, The stability of harmonic maps on Finster manifolds, Houston J. Math. 34 (2008) 1049–1056]) and the same method [S. W. Wei, An extrinsic average variational method, in Recent Developments in Geometry, Contemporary Mathematics, Vol. 101 (American Mathematical Society, Providence, RI, 1989), pp. 55–78].

Open discussion

1982 ◽  
Vol 99 ◽  
pp. 605-613
Author(s):  
P. S. Conti
Keyword(s):  
Buenos Aires ◽  
Large Mass ◽  
List Type ◽  
Common Phenomenon ◽  
Open Discussion ◽  
The Stability ◽  
Ring Nebulae

Conti: One of the main conclusions of the Wolf-Rayet symposium in Buenos Aires was that Wolf-Rayet stars are evolutionary products of massive objects. Some questions:–Do hot helium-rich stars, that are not Wolf-Rayet stars, exist?–What about the stability of helium rich stars of large mass? We know a helium rich star of ∼40 MO. Has the stability something to do with the wind?–Ring nebulae and bubbles : this seems to be a much more common phenomenon than we thought of some years age.–What is the origin of the subtypes? This is important to find a possible matching of scenarios to subtypes.

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