Short-time existence and uniqueness for two-phase harmonic map heat flow with a prescribed moving interface

Let $N^+$ and $N^-$ be disjoint compact smooth submanifolds of a Euclidean space, and let $Λ\subset N^+\times N^-$ be a compact smooth embedded submanifold prescribing admissible pairs of one-sided traces. Given a smooth family of separating hypersurfaces $Γ_t\subset\mathbb T^d$, we study harmonic m