Cohomology Vanishing for Free Boundary $f$-Minimal Submanifolds in Gaussian-Weighted Euclidean Balls

Let $M^n\subset \overline{{\text{B}}_R^{n+k}}\subset {\mathbb{R}}^{n+k}$ be a compact orientable free boundary $f$-minimal submanifold of the Gaussian-weighted Euclidean ball $(\overline{{\text{B}}_R^{n+k}},g_{\mathrm{c}\mathrm{a}\mathrm{n}},e^{-f}\text{dd}V),f(x)=\frac{c}{2}\mathrm{\mid }x{\mathrm{\mid }}^2,c\ge 0.$ We prove a cohomology vanishing theorem under the pointwise pinching condition $\mathrm{\mid }A{\mathrm{\mid }}^2\le \frac{n-p}{R^2},1\le p<n\mathrm{.}$ More precisely, the space of tangential $f$-harmonic $p$-forms vanishes, and hence$H^p(M;\mathbb{R})=0.$ The proof is based on three elementary ingredients in the Gaussian-weighted ball: a weighted Hardy inequality obtained from the identity $\text{divf}(x^T)=n-c\mathrm{\mid }x{\mathrm{\mid }}^2$, a cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction showing that tangential $f$-harmonic forms satisfy the same local absolute-boundary algebra as in the unweighted case. The constant pinching threshold is independent of the Gaussian parameter $c$, and the argument also includes the unweighted case $c=0$; the strict interior positivity comes from the full Hardy–Weitzenböck coefficient rather than from the sign of $c$ alone.