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Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is t...
Let $(M^3,g)$ be a complete, connected, noncompact Riemannian three-manifold without boundary and with nonnegative sectional curvature. We prove that its scalar-curvature integral over geodesic balls, divided by the radius, has a limit. In the one-ended case, \[ \lim_{r\to\infty}\frac1r\int_{B_p(r)}\operatorname{Scal}\...
Let $\Omega\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operato...
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