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Generalized torical band inequalities give precise upper bounds for the widthof compact manifolds with boundary in terms of positive pointwise lower boundsfor scalar curvature, assuming certain topological conditions. We extendseveral incarnations of these results in which pointwise scalar curvaturebounds are replaced with spectral scalar curvature bounds. More precisely, weprove upper bounds for the width in terms of the principal eigenvalue of theoperator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is aconstant. Three separate strategies are employed to obtain distinct resultsholding in different dimensions and under varying hypotheses, namely we utilizespacetime harmonic functions, $\mu$-bubbles, and spinorial Callias operators.In dimension 3, where the strongest result is produced, we are also able totreat open and incomplete manifolds, and establish the appropriate rigiditystatements. Additionally, a version of such spectral torus band inequalities isgiven where tori are replaced with cubes. Finally, as a corollary we generalizeclassical work of Schoen and Yau, on the existence of black holes due toconcentration of matter, to higher dimensions and with alternate measurementsof size.

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