Three circles theorems and Liouville type theorems

We establish three circles theorems for subharmonic functions on Riemannian manifolds with nonnegative Ricci curvature, as well as on gradient shrinking Ricci solitons with scalar curvature bounded from below by $\frac{n-2}{2}$. We also establish a three circiles theorem for holomorphic functions on gradient shrinking Kähler-Ricci solitons with some curvature conditions. As applications, we prove some Liouville type theorems.