Contractible flow of stability conditions via global dimension function

We introduce an analytic method that uses the global dimension function $\mathrm{gldim}$ to produce contractible flows on the space $\mathrm{Stab}\mathcal{D}$ of stability conditions on a triangulated category $\mathcal{D}$. In the case when $\mathcal{D}=\mathcal{D}({\mathbf{S}}^{\lambda })$ is the topological Fukaya category of a graded surface ${\mathbf{S}}^{\lambda }$, we show that ${\mathrm{gldim}}^{-1}(0,y)$ contracts to ${\mathrm{gldim}}^{-1}(0,x)$ for any $1\le x\le y$, provided $(x,y)$ does not contain `critical’ values ${1+w_\partial/m_\partial \mid w_\partial\ge0, \partial\in\partial$\mathbf{S}$^\lambda}$,wherethepair$(m_\partial,w_\partial)$consistsofthenumber$m_\partial$ of marked points and the winding number $w_{\mathrm{\partial }}$ associated to a boundary component $\mathrm{\partial }$ of ${\mathbf{S}}^{\lambda }$. One consequence is that the global dimension of $\mathcal{D}({\mathbf{S}}^{\lambda })$ must be one of these critical values. Besides, we remove the assumptions in Kikuta-Ouchi-Takahashi’s classification result on triangulated categories with global dimension less than 1.