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

Abstract

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 $\overline{\text{Ric}}_{k,φ}(U,V)$, together with the weighted second fundamental form $\widetilde{\mathrm{I\!I}}$, play central roles in controlling the topology of the manifold. Using the theory of $\widetilde{C}(k)$ functions, variational methods, and Morse theory, we prove that by imposing suitable restrictions on these quantities, an $n$-dimensional compact manifold with boundary or a proper open $n$-dimensional manifold is diffeomorphic to an $n$-ball or to a certain CW complex. This shows that such rigidity phenomena is not confined in the unweighted case, but persist in the weighted setting as well. Lastly, we prove a weighted analogue of Frankel's Theorem under positive intermediate weighted Ricci curvature. These results extend classical theorems from the unweighted setting to the weighted case, demonstrating that $\overline{\sec}_φ(U,V)$ and $\overline{\text{Ric}}_{k,φ}(U,V)$ provide effective tools in Riemannian geometry.