{\mathcal{F}} -stability of self-similar solutions to harmonic map heat flow

Inspired by the work of Colding and Minicozzi II: ”Generic mean curvature flow I: generic singularities”, we explore the notion of generic singularities for the harmonic map heat flow. We introduce ${\mathcal{F}}$ -functional and entropy for maps from Euclidean spaces. The critical points of the ${\mathcal{F}}$ -functional are exactly the weakly self-similar solutions to the harmonic map heat flow. We define the notion of ${\mathcal{F}}$ -stability for weakly self-similar solutions. The ${\mathcal{F}}$ -stability can be characterized by the semi-positive definiteness of the Jacobi operator acting on a subspace of variation fields.