Scalar curvature on any closed even-dimensional manifold with nonzero Euler characteristic can be increased by at most an explicit constant, the gap, which is a function of the minimal eigenvalue of the curvature operator of the starting metric.
Scalar curvature rigidity of parabolically convex domains in hyperbolic spaces
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abstract
For a parabolically convex domain $M\subseteq \mathbb{H}^n$, $n\ge 3$, we prove that if $f:(N,\bar g)\to (M,g)$ has nonzero degree, where $N$ is spin with scalar curvature $R_N\ge -n(n-1)$, and if $f|_{\partial N}$ does not increase the distance and the mean curvature, then $N$ is hyperbolic, and $\partial N$ is isometric to $\partial M$. This is a partial generalization of Lott's result \cite{lott2021index} to negative lower bounds of scalar curvature.
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Gap phenomenon for scalar curvature
Scalar curvature on any closed even-dimensional manifold with nonzero Euler characteristic can be increased by at most an explicit constant, the gap, which is a function of the minimal eigenvalue of the curvature operator of the starting metric.