Both bi-harmonic map and $f$-harmonic map have nice physical motivation andapplications. In this paper, by combination of these two harmonic maps, weintroduce and study $f$-bi-harmonic maps as the critical points of the$f$-bi-energy functional $\frac{1}{2}\int_M f|\tau (\phi)|^2dv_{g}$. This classof maps generalizes both concepts of harmonic maps and bi-harmonic maps. Wefirst derive the $f$-biharmonic map equation and then use it to study$f$-bi-harmonicity of some special maps, including conformal maps betweenmanifolds of same dimensions, some product maps between direct product manifoldand singly warped product manifold, some projection maps from and someinclusion maps into a warped product manifold.
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机译：双谐波图和$ f $谐波图都具有良好的物理动力和应用。本文结合这两个谐波图，引入并研究了$ f $-双谐波图作为$ f $-双能量函数$ \ frac {1} {2} \ int_M f |的临界点。 \ tau（\ phi）| ^ 2dv_ {g} $。此类地图概括了谐波图和双谐波图的概念。我们首先导出$ f $-双调和映射方程，然后用它来研究某些特殊映射的$ f $-双调和，包括相同尺寸的流形之间的共形映射，直接乘积流形和单扭曲乘积流形之间的某些乘积图，一些投影映射和包含映射到扭曲的产品流形。
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