Gap Phenomenon for Yamabe Type Problems of $M^m\times T^{n-m}$

We prove that if $(M^m,h)$ is a Yamabe metric, then the product metric $h+g_{\mathrm{f}\mathrm{l}\mathrm{a}\mathrm{t}}$ on $M^m\times T^{n-m}$ is also a Yamabe metric whenever the flat torus $T^{n-m}$ is sufficiently small. This generalizes earlier results for $S^1\times S^{n-1}$. Our method extends to the study of Type~I and Type~II Yamabe constants, $Q$-curvature problems, and an isoperimetric-ratio type problem.