Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds

1989 ◽  
Vol 7 (1) ◽  
pp. 47-57 ◽  
Author(s):  
Seiki Nishikawa ◽  
Philippe Tondeur
Keyword(s):  
Complete Manifolds ◽  
Harmonic Foliations ◽  
Infinitesimal Automorphisms

scholarly journals On VT-harmonic maps

2019 ◽  
Vol 57 (1) ◽  
pp. 71-94 ◽  
Author(s):  
Qun Chen ◽  
Jürgen Jost ◽  
Hongbing Qiu
Keyword(s):  
Maximum Principle ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Existence Results ◽  
Gradient Estimates ◽  
Hermitian Manifolds ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Levi Civita Connection ◽  
Complete Manifolds

Abstract 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 Jäger–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.

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