Tilting-completion for gentle algebras

It is demonstrated that any almost-tilting module over a gentle algebra is indeed partial-tilting, meaning it can be completed as a tilting module. Furthermore, such a module has at most $2n$ possible complements, thereby confirming a (modified) conjecture of Happel for the case of gentle algebras. Additionally, for any $n\geq 3$ and $1\leq m \leq n-2$, there always exists a (connected) gentle algebra with rank $n$ and a pre-tilting module of rank $m$ which is not partial-tilting. The tool we use is the surface model associated with the module category of a gentle algebra. The main technique is an induction process involving surface cuts, which is hoped to be beneficial for other applications as well.