标题： 谱环带不等式和Schoen-Yau黑洞存在定理的推广 显示英文标题

标题： Spectral Torical Band Inequalities and Generalizations of the Schoen-Yau Black Hole Existence Theorem

摘要： 广义环带不等式给出了在给定标量曲率的点态正下界的情况下，具有边界的紧致流形宽度的精确上界，假设满足某些拓扑条件。 我们扩展了这些结果的几种形式，在这些形式中，点态标量曲率界限被谱标量曲率界限所取代。 更准确地说，我们证明了宽度的上界，该上界以算子$-\Delta +cR$的主特征值为基准，其中$R$表示标量曲率，$c>0$是一个常数。 为了在不同维度和不同假设下获得不同的结果，采用了三种独立的策略，即我们利用了时空调和函数、$\mu$-气泡和旋量 Callias 算子。 在产生最强结果的三维情况下，我们还能够处理开流形和不完全流形，并建立适当的刚性陈述。 此外，还给出了一种这样的谱环带不等式的版本，其中环面被立方体所取代。 最后，作为推论，我们将 Schoen 和 Yau 关于由于物质集中而存在黑洞的经典工作推广到高维，并使用了其他尺寸测量方法。 显示英文摘要

摘要： Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, $\mu$-bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary we generalize classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size.