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In this paper we prove classification results to elliptic fully nonlinearconformal equations on certain subdomains of the sphere with prescribedconstant mean curvature on its boundary. Such subdomains are the hemisphere (ora geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribedconstant mean curvature on its boundary, and annular domains with minimalboundary. Our results extend the classifications of Escobar in \cite{E0} when$n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.

Uniqueness Theorems for fully nonlinear conformal equations on subdomains of the sphere