Optimal connectivity results for spheres in the curve graph of low and medium complexity surfaces

Answering a question of Wright, we show that spheres of any radius are always connected in the curve graph of surfaces ${\mathrm{\Sigma }}_{2,0},{\mathrm{\Sigma }}_{1,3},$ and ${\mathrm{\Sigma }}_{0,6}$, and the union of two consecutive spheres is always connected for ${\mathrm{\Sigma }}_{0,5}$ and ${\mathrm{\Sigma }}_{1,2}$. We also classify the connected components of spheres of radius 2 in the curve graph of ${\mathrm{\Sigma }}_{0,5}$ and ${\mathrm{\Sigma }}_{1,2}$.