标题： 程的度量测度空间上的特征值比较及其应用 显示英文标题

标题： Cheng's eigenvalue comparison on metric measure spaces and applications

摘要： 使用局部化技术，我们证明了在本质上不分支的$\mathsf{CD}^{\star}(K,N)$空间中的度量球的第一个狄利克雷特征值的精确上界。 这将 Cheng 的一个著名结果扩展到了通过最优传输满足合成意义下里奇曲率下界的度量测度空间的非光滑情形。 提供了$\mathsf{RCD}^{\star}(K,N)$空间的刚性与稳定性陈述；即使对于光滑黎曼流形，这种稳定性似乎也是新的。 然后我们提出了一些数学和物理应用：在前者中，我们得到了本质上不分支的$\mathsf{CD}^{\star}(K,N)$空间中$j^{th}$拉普拉斯特征值的上界以及非紧致$\mathsf{RCD}^{\star}(K,N)$空间中本质谱的界限；在后者中，特征值界限对应于高维引力理论的一般扭曲紧化周围自旋-2 卡鲁扎-克莱因激发态质量的一般上界。 显示英文摘要

摘要： Using the localization technique, we prove a sharp upper bound on the first Dirichlet eigenvalue of metric balls in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces. This extends a celebrated result of Cheng to the non-smooth setting of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense, via optimal transport. Rigidity and stability statements are provided for $\mathsf{RCD}^{\star}(K,N)$ spaces; the stability seems to be new even for smooth Riemannian manifolds. We then present some mathematical and physical applications: in the former, we obtain an upper bound on the $j^{th}$ Neumann eigenvalue in essentially non-branching $\mathsf{CD}^{\star}(K,N)$ spaces and a bound on the essential spectrum in non-compact $\mathsf{RCD}^{\star}(K,N)$ spaces; in the latter, the eigenvalue bounds correspond to general upper bounds on the masses of the spin-2 Kaluza-Klein excitations around general warped compactifications of higher-dimensional theories of gravity.