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In this paper we prove rigidity results for two-dimensional, closed,immersed, non-necessarily convex, self-similar solutions of a wide class offully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similarsolutions are the round spheres centered at the origin provided it has genuszero and satisfies a suitable upper pinching estimate for the Gaussiancurvature. As applications, we obtain rigidity results for the round sphere asthe only closed, immersed, genus zero, self-similar solution of several wellknown flows, as the flow of the powers of mean curvature, the harmonic meancurvature flow and the $\alpha$-Gaussian curvature flow for $\alpha\in(0,1/4)$.We remark that our result does not assume any embeddedness condition.

Hopf type theorems for self-similar solutions of curvature flows in $\mathbb{R}^3$