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 The second variation formula for exponentially harmonic maps

1999 ◽  
Vol 59 (3) ◽  
pp. 509-514 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Compact Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Sharp Contrast ◽  
Second Variation ◽  
Variation Formula ◽  
Totally Geodesic ◽  
The Second Variation

We derive the formula in the title and deduce some consequences. For example we show that the identity map from any compact manifold to itself is always stable as an exponentially harmonic map. This is in sharp contrast to the harmonic or p-harmonic cases where many such identity maps are unstable. We also prove that an isometric and totally geodesic immersion of Sm into Sn is an unstable exponentially harmonic map if m ≠ n and is a stable exponentially harmonic map if m = n.

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].

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.

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 conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

1982 ◽  
Vol 91 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. C. J. Sealey
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Differential Forms ◽  
Finite Energy ◽  
Energy Functional ◽  
Conformal Transformations ◽  
Second Variation ◽  
Yang Mills

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

scholarly journals Exponentially Harmonic Maps into Spheres

Axioms ◽  
2018 ◽  
Vol 7 (4) ◽  
pp. 88
Author(s):  
Sorin Dragomir ◽  
Francesco Esposito
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Codimension Two ◽  
Geodesic Submanifold

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m ⊂ R m + 1 . Given a codimension two totally geodesic submanifold Σ ⊂ S m , we show that every nonconstant exponentially harmonic map ϕ : M → S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) ∩ Σ ≠ ∅ .

Complete Riemannian Manifold with Curvature Bounded from Below

2012 ◽  
Vol 182-183 ◽  
pp. 1225-1229
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao
Keyword(s):  
Riemannian Manifold ◽  
Complete Riemannian Manifold ◽  
Second Variation ◽  
Variation Formula ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
First Variation ◽  
Geodesic Submanifold ◽  
The First Variation ◽  
The Second Variation

In this paper, we study a complete -Riemannian manifold whose curvature bounded from below. Let be a compact totally geodesic submanifold of . Then, for any , we can make use of the first variation formula and the second variation formula of distance to prove that is bounded.

Harmonic G-structures

2009 ◽  
Vol 146 (2) ◽  
pp. 435-459 ◽  
Author(s):  
J. C. GONZÁLEZ–DÁVILA ◽  
F. MARTÍN CABRERA
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Second Variation ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Intrinsic Torsion ◽  
First And Second Variation

AbstractFor closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold (M, 〈⋅, ⋅〉), where G-structures are considered as sections of the quotient bundle (M)/G. We deduce the corresponding first and second variation formulae and the characterising conditions for critical points by means of tools closely related to the study of G-structures. In this direction, we show the rôle in the energy functional played by the intrinsic torsion of the G-structure. Moreover, we analyse the particular case G=U(n) for 2n-dimensional manifolds. This leads to the study of harmonic almost Hermitian manifolds and harmonic maps from M into (M)/U(n).

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals Autoparallel Deviation in the Geometry of Lyra

1960 ◽  
Vol 3 (3) ◽  
pp. 263-271 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Calculus Of Variations ◽  
Finsler Manifold ◽  
Geometrical Approach ◽  
Second Variation ◽  
Geodesic Deviation ◽  
The Difference ◽  
The Second Variation

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

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