  In this article, we study the uniqueness problem for the generalized gauss
maps of minimal surfaces (with the same base) immersed in $\mathbb R^{n+1}$
which have the same inverse image of some hypersurfaces in a projective
subvariety $V\subset\mathbb P^n(\mathbb C)$. As we know, this is the first time
the unicity of generalized gauss maps on minimal surfaces sharing hypersurfaces
in a projective varieties is studied. Our results generalize and improve the
previous results in this field.
