2014
DOI: 10.1007/s12220-014-9519-9
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A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

Abstract: In this note we prove that the maximum principle of Jäger-Kaul for harmonic maps holds for a more general class of maps, V -harmonic maps. This includes Hermitian harmonic maps (Jost and Yau, Acta Math 170:221-254, 1993), Weyl harmonic maps (Kokarev, Proc Lond Math Soc 99:168-194, 2009), affine harmonic maps Jost and Simsir (Analysis (Munich) 29: [185][186][187][188][189][190][191][192][193][194][195][196][197] 2009), and Finsler maps from a Finsler manifold into a Riemannian manifold. With this maximum princi… Show more

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Cited by 31 publications
(20 citation statements)
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References 39 publications
(57 reference statements)
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“…The notion of V-harmonic maps was introduced in [5]. It includes the Hermitian harmonic maps introduced and studied in [17], the Weyl harmonic maps from a Weyl manifold into a Riemannian manifold [18], the affine harmonic maps mapping from an affine manifold into a Riemannian manifold [15], [16], and harmonic maps from a Finsler manifold into a Riemannian manifold [2], [12], [30], [33] and [31], see [5] for explanation of these relations. Another interesting special case is when V is a gradient vector field, i.e., V = ∇f for some function f : M → R, then (1.1) takes the form In this case, (1.2) is of divergence form.…”
Section: Introductionmentioning
confidence: 99%
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“…The notion of V-harmonic maps was introduced in [5]. It includes the Hermitian harmonic maps introduced and studied in [17], the Weyl harmonic maps from a Weyl manifold into a Riemannian manifold [18], the affine harmonic maps mapping from an affine manifold into a Riemannian manifold [15], [16], and harmonic maps from a Finsler manifold into a Riemannian manifold [2], [12], [30], [33] and [31], see [5] for explanation of these relations. Another interesting special case is when V is a gradient vector field, i.e., V = ∇f for some function f : M → R, then (1.1) takes the form In this case, (1.2) is of divergence form.…”
Section: Introductionmentioning
confidence: 99%
“…Therefore many useful tools developed in the theory of harmonic maps are no longer available, and the analysis here is more difficult. In [5], by combining a maximum principle of Jäger-Kaul type [13] with the continuity method, the authors established an existence and uniqueness theorem for V -harmonic maps from compact Riemannian manifolds with boundary into a regular ball. Here, and in the sequel, a regular ball B R (p) is a distance ball in X that is disjoint from the cut-locus of its center p and with radius R <…”
Section: Introductionmentioning
confidence: 99%
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“…该文献发展了关于 Hermite 调和映照的分析理论, 在某个必要的拓扑 非平凡条件下, 得到了当目标流形是非正截面曲率时解的存在性. 进一步的结果被文献 [15,[19][20][21][22][23] 等 得到. 关于在 Hermite 几何中的刚性理论方面的应用, 参见文献 [18].…”
Section: Hermite 调和映照unclassified
“…An important generalization is a diffusion operator (1.1) ∆ V := ∆ + V, ∇ on a Riemannian manifold (M, g) of dimension m, where ∇ and ∆ are respectively the Levi-Civita connection and Beltrami-Laplace operator of g, and where V is a smooth vector field on M. This operator is also a special case of V -harmonic map introduced in [11]. As in [4,10], we introduce Bakey-Emery Ricci tensor fields is exactly the Ricci soliton equation, which is one-to-one corresponding to a selfsimilar solution of Ricci flow (see, [13] It is easy to see that the scalar curvature of g cs is 4/(1+x 2 +y 2 ) and hence the cigar soliton is not Ricci-flat.…”
Section: Introductionmentioning
confidence: 99%