VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jager–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

中文翻译：

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在 VT 谐波图上

VT 谐波图概括了标准谐波图，涉及域和目标的结构。这些可以是具有自然连接的流形，而不是黎曼几何的列维-奇维塔连接，如厄米流形、仿射流形或外尔流形。标准谐波映射半线性椭圆系统由来自域上矢量场 V 的项和来自目标上的 2-张量 T 的另一项增强。事实上，这种几何结构还包括其他几何定义的图，例如磁谐波图。在本文中，我们使用 PDE 工具处理 VT 谐波映射及其抛物线类似物。我们为这些地图建立了 Jager-Kaul 类型最大值原理。使用这个最大值原理，我们证明了 VT 谐波映射的狄利克雷问题的存在定理。作为应用程序，我们获得了 Weyl/affine/Hermitian 流形之间的 Weyl/affine/Hermitian 谐波映射的结果，以及来自二维域的磁谐波映射的结果。我们还从非紧实的完全流形导出梯度估计并获得此类映射的存在结果。