On the blow up phenomenon for the $L^2$-critical focusing Hartree equation in ${\mathbb{R}}^4$

We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1({\mathbb{R}}^4)$ data and $L^2({\mathbb{R}}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.