具有加权凸边界的紧流形和正常流形上的正加权曲率
Positive Weighted Curvatures on Compact Manifolds with Weighted Convex Boundary and Proper Manifolds
AI总结：
该论文研究加权黎曼流形上的正加权曲率，证明在加权截面曲率和中间加权 Ricci 曲率限制下，紧流形或正常流形微分同胚于球或 CW 复形，并推广了 Frankel 定理。
AI中文摘要：
给定一个配备正密度函数 $f = e^{-\varphi}$ 的黎曼流形 $(M,g)$，我们研究加权截面曲率 $\overline{\sec}_\varphi(U,V)$ 和第 $k$ 中间加权 Ricci 曲率 $\overline{\text{Ric}}_{k,\varphi}(U,V)$。我们证明 $\overline{\sec}_\varphi(U,V)$ 和 $\overline{\text{Ric}}_{k,\varphi}(U,V)$ 连同加权第二基本形式 $\widetilde{\mathrm{I\\!I}}$ 在控制流形的拓扑中起核心作用。利用 $\widetilde{C}(k)$ 函数理论、变分方法和 Morse 理论，我们证明通过对这些量施加适当的限制，一个具有边界的 $n$ 维紧流形或一个正常开的 $n$ 维流形微分同胚于一个 $n$ 维球或某个 CW 复形。这表明这种刚性现象不仅限于无加权情形，而且在加权情形中同样存在。最后，我们在正中间加权 Ricci 曲率下证明了 Frankel 定理的加权类比。这些结果将无加权情形下的经典定理推广到加权情形，表明 $\overline{\sec}_\varphi(U,V)$ 和 $\overline{\text{Ric}}_{k,\varphi}(U,V)$ 在黎曼几何中提供了有效的工具。
英文摘要：
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.
