Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds

Given a Riemannian manifold $(M,g)$ equipped with a positive density function $f = e^{-φ}$, we study the weighted sectional curvature $\overline{\sec}_φ(U,V)$ and the $k$-th intermediate weighted Ricci curvature $\overline{\text{Ric}}_{k,φ}(U,V)$. We show that $\overline{\sec}_φ(U,V)$ and $\overline