Mixed Radial Volume Comparison under Spectral Ricci Bounds

Abstract

Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-α\frac{Δ_g u}{u}g\ge (n-1)κg$. We introduce mixed radial balls associated with the conformal metric $u^{2α}g$. 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 $\mathbb{R}^4$.