Minimal weights of Hilbert modular forms in characteristic p

We consider mod p HIlbert modular forms associated to a totally real field of degree d in which p is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a d-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren-Oort stratification on mod p HIlbert modular varieties established by Goren and Oort, and Tian and Xiao.