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 principle we establish the existence of V -harmonic maps into regular balls.

Existence and Liouville theorems for V -harmonic maps from complete manifolds

“…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.…”

“…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 <…”

“…[5]). Let (M, g) be a compact Riemannian manifold with boundary ∂M and (X, h) a complete Riemannian manifold without boundary.…”

Harmonic maps and their generalizations

“…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].…”

Li–Yau–Hamilton estimates and Bakry–Emery–Ricci curvature

“…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.…”