Mapping class group orbit closures for Deroin–Tholozan representations

Figure 1.  Left: a $ 4 $-punctured sphere with two curves $ b $ and $ d $ in torso configuration. Right: a DT component (which is symplectically a sphere) and the Lagrangian submanifold cut out by $ \{\beta, \delta\} = 0 $

Figure 2.  The standard pants decomposition of $ \Sigma $ associated to the generators $ (c_1, \ldots, c_n) $

Figure 3.  Example of a triangle chain in the case $ n = 6 $

Figure 4.  Visualisation of the action-angle $ \beta_i $ and $ \gamma_i $ on a triangle chain in the case $ n = 6 $

Figure 5.  The curves $ b $, $ d $, and $ e $

Figure 6.  In black: the $ \mathscr{B} $-triangle chain of $ [\rho] $. In mauve: the first triangle in the $ \mathscr{D} $-triangle chain of $ [\rho] $. In purple: the first triangle in the $ \mathscr{E} $-triangle chain of $ [\rho] $

Figure 7.  A DT component isomorphic to a sphere and the locus $ \{\beta, \gamma\} = 0 $

Figure 8.  The curves $ b_i $, $ d_i $, and $ e_i $

Figure 9.  In black: the $ i^\mathrm{th} $ and $ (i+1)^\mathrm{th} $ triangles in the $ \mathscr{B} $-triangle chain of $ [\rho] $. In mauve: the first triangle in the $ \mathscr{D}_i $-triangle chain of $ [\rho] $. In purple: the $ i^\mathrm{th} $ triangle in the $ \mathscr{E}_i $-triangle chain of $ [\rho] $

Figure 10.  The $ \mathscr{B} $-triangle chain of $ \phi $

Figure 11.  The sub-sphere $ \overline{\Sigma} $

Figure 12.  The $ \mathscr{B} $-triangle chain of a point in $ A_j^{\pi} $

Figure 13.  The sub-spheres $ \Sigma_1 $ and $ \Sigma_2 $

Figure 14.  The $ \mathscr{D}_i $-triangle chain of $ [\phi] $ and the angle $ \gamma_1^{\mathscr{D}_i} $

Figure 15.  A $ \mathscr{D}'_i $-triangle chain and the angle $ \gamma_1^{\mathscr{D}'_i} $