Let $(M^n,g)$ be a complete Riemannian manifold with $n\geq 2$ and $\operatorname{Ric}_g\ge (n-1)k g$. Fix $p\in M$ and suppose that for some $0<r<R$ with $r,R\in I_k$,

\begin{align*} \frac{\operatorname{vol}_g(B(p,R))}{V_k^n(R)}= \frac{\operatorname{vol}_g(B(p,r))}{V_k^n(r)}. \end{align*}

Then the radial density defect used in the Bishop-Gromov proof vanishes for $\sigma_p\otimes\mathcal{L}^1$-almost every pair $(\theta,\rho)$ with $\theta\in S_pM$, $r<\rho<R$, and $\rho<\tau(\theta)$, where $\sigma_p$ is the Riemannian measure on the unit sphere $S_pM\subset T_pM$ and $\tau:S_pM\to(0,\infty]$ is the cut-time function. In particular, the geodesic sphere mean curvature agrees with the model value almost everywhere in the polar-coordinate region of that annulus.

Assume in addition that the annulus is free of cut points, meaning $\tau(\theta)>R$ for every $\theta\in S_pM$. For each $\theta\in S_pM$, let

\begin{align*} \gamma_\theta:[0,R)&\to M,\qquad \gamma_\theta(\rho)=\exp_p(\rho\theta), \end{align*}

be the unit-speed radial geodesic, and let

\begin{align*} S_\theta(\rho):\dot\gamma_\theta(\rho)^\perp&\to\dot\gamma_\theta(\rho)^\perp \end{align*}

be the radial shape operator of the geodesic sphere through $\gamma_\theta(\rho)$. If for every radial geodesic the smooth equality conditions in the Riccati trace estimate hold, namely the Cauchy-Schwarz equality case for $S_\theta(\rho)$ and the radial Ricci equality

\begin{align*} \operatorname{Ric}_g(\dot\gamma_\theta(\rho),\dot\gamma_\theta(\rho))=(n-1)k \end{align*}

hold for all $\rho\in(r,R)$, then the almost-everywhere mean-curvature equality upgrades to

\begin{align*} S_\theta(\rho)=\operatorname{ct}_k(\rho)\operatorname{Id}_{\dot\gamma_\theta(\rho)^\perp} \end{align*}

for all $\theta\in S_pM$ and all $\rho\in(r,R)$, and the polar metric has the corresponding model radial warping. A full round model-annulus isometry requires the angular metric data at one radius to match the model angular metric as well.