Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\geq (n-1)\kappa g$. 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 $\mathbb R^4$.
This paper develops radial volume comparison under a spectral lower bound for Ricci curvature. The mixed radial construction separates the conformal geometry used to choose rays from the weights used to truncate and measure them.
Sharp comparison
Under the spectral inequality $\operatorname{Ric}_g-\alpha(\Delta_g u/u)g\geq(n-1)\kappa g$, the weighted volume of the $g$-arclength mixed radial domain satisfies a scale-invariant comparison with the corresponding space-form volume.
Polynomial growth
For an explicit range of radial and measure exponents, mixed radial balls have polynomial weighted-volume growth. The optimal exponent can attain the Euclidean value $n$.
Applications
The comparison gives radial proofs of the spectral Bonnet--Myers and sharp volume theorem of Antonelli--Xu, together with a short volume-growth proof that every complete two-sided stable minimal immersion $M^3\to\mathbb R^4$ is flat.
The argument begins with an exact Riccati identity for the weighted radial Jacobian along geodesics of the conformal metric $h_\alpha=u^{2\alpha}g$. Taking the $(n-1)$-st root of this Jacobian converts the identity into a one-dimensional Sturm comparison and yields the space-form estimate. For general mixed radial balls, the logarithmic derivatives of the Jacobian and the spectral weight are instead treated as coupled variables; a quadratic estimate and a Bellman-type correction produce the polynomial-growth monotonicity formula. The positive-curvature case also gives an optimal diameter bound among smooth closed examples.
My research interests include Geometric Analysis and Partial Differential Equations.