Tight contact structures on some bounded Seifert manifolds with minimal convex boundary

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0, 1)$ and $r_1,r_2\in [1/2,1)$; (3) $s\in[1, 2)$ and $r_1,r_2\in(0,1/2)$; (4) $s=\infty$ and $r_1=r_2=1/2$. We also classify positive tight contact structures, up to isotopy fixing the boundary, on $M(D^2;1/2,1/2)$ with minimal convex boundary of arbitrary slope and Giroux torsion greater than 0 along the boundary.

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