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In this paper we examine analytically the large-$N$ gap equation and itssolution for the $2D$ $\mathbb{CP}^{N-1}$ sigma model defined on a Euclideanspacetime torus of arbitrary shape and size ($L, \beta)$, $\beta$ being theinverse temperature. We find that the system has a unique homogeneous phase,with the $\mathbb{CP}^{N-1}$ fields $n_i$ acquiring a dynamically generatedmass $\langle\lambda\rangle\ge\Lambda^2$ (analogous to the mass gap of $SU(N)$Yang-Mills theory in $4D$), for any $\beta$ and $L$. Several related topics inthe recent literature are discussed. One concerns the possibility, which turnsout to be excluded according to our analysis, of a "Higgs-like" - ordeconfinement - phase at small $L$ and at zero temperature. Another topicsinvolves "soliton-like (inhomogeneous) solutions of the generalized gapequation, which we do not find. A related question concerns a possibleinstability of the standard $\mathbb{CP}^{N-1}$ vacuum on ${\mathbb{R}}^{2}$,which is shown not to occur. In all cases, the difference in the conclusionscan be traced to the existence of certain zeromodes and their proper treatment.The $\mathbb{CP}^{N-1}$ model with twisted boundary conditions is alsoanalyzed. The $\theta$ dependence and different limits involving $N$, $\beta$and $L$ are briefly discussed.

Large-$N$ $\mathbb{CP}^{N-1}$ sigma model on a Euclidean torus: uniqueness and stability of the vacuum