Relative stability conditions on Fukaya categories of surfaces

It is shown that there a useful notion of relative Bridgeland stability condition on the partially wrapped Fukaya category marked surface, with respect to some part surface’s boundary. This construction has nice functorial properties, obeying cutting and gluing, reducing calculation conditions any fully stopped surface into three types base cases. Calculations these cases shows every such categories can be described by flat surfaces. In other words, when stopped, map constructed Haiden–Katzarkov–Kontsevich from moduli surfaces space global homeomorphism.