arXiv:2610.36091v1 Announce Type: new Abstract: In this paper, we establish spectral Bakry-\'{E}mery $\Gamma_2$ criteria and their modified counterparts for the $V$-Laplacian on closed Riemannian manifolds equipped with evolving weighted measures. The estimates yield jointly optimized spectral bounds and sharp endpoint constants. When the measures evolve according to the Fokker-Planck equation, these estimates yield monotonicity formulas for the Fisher information associated with Shannon and Tsallis entropies, together with quantitative propagation of Poincar\'e, logarithmic Sobolev, and nonlinear interpolation inequalities. In the stationary reversible setting, our results specialize to the Witten Laplacian and recover the unweighted Cai-Wang estimates \cite{Cai}.
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