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We study emergent behaviors of the Lohe hermitian sphere(LHS) model which isan aggregation model on ${\mathbb C}^d$. The LHS model is a complex analog ofthe Lohe sphere model on ${\mathbb R}^d$, and hermitian spheres are invariantsets for the LHS dynamics. For the derivation of the LHS model, we use atop-down approach, namely a reduction from a high-rank aggregation model,"$\textit{the Lohe tensor model}$." The Lohe tensor model is a first-orderaggregation model on the space of tensors with the same rank and sizes, and itwas first proposed by the authors in a recent work \cite{H-P}. In this work, westudy how the LHS model appears as a special case of the Lohe tensor model andfor the proposed model, we provide a cross-ratio like conserved quantity, asufficient framework for the complete aggregation and a uniform$\ell^p$-stability estimate with respect to initial data.

From the Lohe tensor model to the Lohe Hermitian sphere model and emergent dynamics