Mixed Radial Volume Comparison under Spectral Ricci Bounds

Let $(M^n,g)$ be complete, let $u>0$, and assume ${\mathrm{Ric}}_g-\alpha \frac{{\mathrm{\Delta }}_{g}u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha }g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet–Myers and sharp volume theorem of Antonelli–Xu, and a short volume growth derivation of the stable Bernstein theorem in ${\mathbb{R}}^4$.