[proofplan] We write the Riemannian volume measure in geodesic polar coordinates about $p$. Laplacian comparison gives a differential inequality for the logarithmic radial derivative of the polar Jacobian before the cut time, and comparison with the model function $s_k(t)^{n-1}$ shows that each radial density ratio is nonincreasing. Integrating this pointwise monotonicity over the unit tangent directions, and using the polar formula for the derivative of ball volume at regular radii, gives the asserted almost-everywhere monotonicity of the distance-sphere area ratio. [/proofplan]

[step:Introduce the polar Jacobian and the cut time] Let $U_pM := \{v \in T_pM : g_p(v,v)=1\}$ be the unit tangent sphere at $p$, equipped with the Riemannian hypersurface measure $\sigma_p := \mathcal{H}^{n-1}\big|_{U_pM}$. For $\theta \in U_pM$, let \begin{align*} \gamma_\theta : [0,\infty) &\to M \\ t &\mapsto \exp_p(t\theta) \end{align*} be the unit-speed geodesic starting at $p$ in direction $\theta$. Let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}$. Define the admissible model interval $I_k$ by \begin{align*} I_k := \begin{cases} (0,\infty), & k \leq 0,\\ (0,\pi/\sqrt{k}), & k>0. \end{cases} \end{align*} Define the cut time \begin{align*} c: U_pM &\to (0,\infty] \\ \theta &\mapsto \sup\{t>0 : \gamma_\theta|_{[0,t]} \text{ is minimizing}\}. \end{align*} The cut locus of $p$ is the set \begin{align*} \operatorname{Cut}(p) := \{\exp_p(c(\theta)\theta) : \theta \in U_pM,\ c(\theta)<\infty\}. \end{align*} By the standard cut-locus and geodesic polar-coordinate theorem for complete Riemannian manifolds, applied to the complete manifold $(M,g)$ and the basepoint $p$, the cut time $c:U_pM\to(0,\infty]$ is Borel measurable and $\operatorname{Cut}(p)$ has zero Riemannian volume. The same theorem supplies a smooth positive polar Jacobian density \begin{align*} J: \{(t,\theta)\in (0,\infty)\times U_pM : 0<t<c(\theta)\} &\to (0,\infty) \\ (t,\theta) &\mapsto J(t,\theta), \end{align*} characterized by the identity that, under the map \begin{align*} \Phi: \{(t,\theta): \theta \in U_pM,\ 0<t<c(\theta)\} &\to M \setminus (\{p\}\cup \operatorname{Cut}(p)) \\ (t,\theta) &\mapsto \exp_p(t\theta), \end{align*} the Riemannian volume measure decomposes as \begin{align*} d\operatorname{Vol}_g(\Phi(t,\theta)) = J(t,\theta)\, d\mathcal{L}^1(t)\, d\sigma_p(\theta). \end{align*} The geodesic polar-coordinate formula with cut locus, a standard consequence of the fact that $\operatorname{Cut}(p)$ has zero Riemannian volume, gives \begin{align*} V_p(t) = \int_{U_pM} \int_0^{\min\{t,c(\theta)\}} J(s,\theta)\, d\mathcal{L}^1(s)\, d\sigma_p(\theta) \end{align*} for every $t \in I_k$. [/step]

[step:Compare each radial Jacobian density with the model density]Fix $\theta \in U_pM$ and $0<t<c(\theta)$ with $t\in I_k$. Define the model warping function \begin{align*} s_k: I_k &\to (0,\infty) \\ t &\mapsto \begin{cases} \frac{1}{\sqrt{k}}\sin(\sqrt{k}t), & k>0,\\ t, & k=0,\\ \frac{1}{\sqrt{-k}}\sinh(\sqrt{-k}t), & k<0. \end{cases} \end{align*} By this definition, $s_k(t)>0$ for every $t\in I_k$, so logarithms and division by $s_k(t)^{n-1}$ are legitimate. On $M \setminus (\{p\}\cup \operatorname{Cut}(p))$, let \begin{align*} \rho: M \setminus \{p\} &\to (0,\infty) \\ q &\mapsto d_g(p,q) \end{align*} be the distance function from $p$. Along the radial geodesic $\gamma_\theta$, the polar Jacobian satisfies \begin{align*} \frac{\partial}{\partial t}\log J(t,\theta) = \Delta_g \rho(\gamma_\theta(t)). \end{align*} By the standard [Laplacian Comparison Theorem](/theorems/5360) for the Distance Function, applied on $M\setminus(\{p\}\cup\operatorname{Cut}(p))$ using the lower Ricci bound $\operatorname{Ric}_g \geq (n-1)k g$, for every $0<t<c(\theta)$ with $t \in I_k$, \begin{align*} \Delta_g \rho(\gamma_\theta(t)) \leq (n-1)\frac{s_k'(t)}{s_k(t)}. \end{align*} Therefore \begin{align*} \frac{\partial}{\partial t}\log \left(\frac{J(t,\theta)}{s_k(t)^{n-1}}\right) = \frac{\partial}{\partial t}\log J(t,\theta) - (n-1)\frac{s_k'(t)}{s_k(t)} \leq 0. \end{align*} Thus, for each fixed $\theta \in U_pM$, the function \begin{align*} (0,c(\theta))\cap I_k &\to (0,\infty) \\ t &\mapsto \frac{J(t,\theta)}{s_k(t)^{n-1}} \end{align*} is nonincreasing. Equivalently, whenever $0<t_1<t_2<c(\theta)$ and $t_1,t_2\in I_k$, \begin{align*} \frac{J(t_2,\theta)}{s_k(t_2)^{n-1}} \leq \frac{J(t_1,\theta)}{s_k(t_1)^{n-1}}. \end{align*}[/step]

[guided]The purpose of this step is to turn Ricci curvature information into a one-dimensional monotonicity statement along each minimizing radial geodesic. Fix a direction $\theta \in U_pM$ and consider the geodesic \begin{align*} \gamma_\theta : [0,\infty) &\to M \\ t &\mapsto \exp_p(t\theta). \end{align*} As long as $0<t<c(\theta)$, the point $\gamma_\theta(t)$ is outside the cut locus of $p$, so geodesic polar coordinates are smooth there and the distance function \begin{align*} \rho: M \setminus \{p\} &\to (0,\infty) \\ q &\mapsto d_g(p,q) \end{align*} is smooth at $\gamma_\theta(t)$. The polar Jacobian $J(t,\theta)$ measures the infinitesimal volume expansion of the exponential map in the angular directions. Its logarithmic radial derivative is the mean curvature of the distance sphere, equivalently the Laplacian of the distance function: \begin{align*} \frac{\partial}{\partial t}\log J(t,\theta) = \Delta_g \rho(\gamma_\theta(t)). \end{align*} This identity is the bridge between volume density and Laplacian comparison. Now apply the standard Laplacian Comparison Theorem for the Distance Function under the hypothesis \begin{align*} \operatorname{Ric}_g(v,v) \geq (n-1)k\,g(v,v) \end{align*} for every $v \in TM$. The theorem applies on $M \setminus (\{p\}\cup \operatorname{Cut}(p))$, and therefore along $\gamma_\theta(t)$ for $0<t<c(\theta)$. It gives \begin{align*} \Delta_g \rho(\gamma_\theta(t)) \leq (n-1)\frac{s_k'(t)}{s_k(t)}. \end{align*} The model sphere density in the simply [connected space](/page/Connected%20Space) form is $s_k(t)^{n-1}$, and its logarithmic derivative is \begin{align*} \frac{\partial}{\partial t}\log(s_k(t)^{n-1}) = (n-1)\frac{s_k'(t)}{s_k(t)}. \end{align*} Subtracting the model logarithmic derivative from the manifold logarithmic derivative gives \begin{align*} \frac{\partial}{\partial t}\log \left(\frac{J(t,\theta)}{s_k(t)^{n-1}}\right) = \frac{\partial}{\partial t}\log J(t,\theta) - (n-1)\frac{s_k'(t)}{s_k(t)} \leq 0. \end{align*} Hence the ratio \begin{align*} \frac{J(t,\theta)}{s_k(t)^{n-1}} \end{align*} cannot increase as $t$ increases before the cut time. This is the radial density comparison that will be integrated over all directions.[/guided]

[step:Construct the full-measure set where volume derivative, polar density, and sphere area agree] Define the ball-volume function \begin{align*} V_p: I_k &\to [0,\infty) \\ t &\mapsto \operatorname{Vol}_g(B(p,t)) \end{align*} and define \begin{align*} b: I_k &\to [0,\infty] \\ t &\mapsto \int_{\{\theta \in U_pM : c(\theta)>t\}} J(t,\theta)\, d\sigma_p(\theta). \end{align*} The Borel measurability of $c$ and the smoothness of $J$ on $\{(t,\theta):0<t<c(\theta)\}$ make the integrand $\mathbb{1}_{\{c>t\}}(\theta)J(t,\theta)$ measurable for each $t\in I_k$. The polar-coordinate formula above writes $V_p(t)$ as an integral of the non-negative measurable function $\mathbb{1}_{\{0<s<\min\{t,c(\theta)\}\}}J(s,\theta)$ over $(0,t)\times U_pM$. Applying Tonelli's theorem to this non-negative integrand gives the explicit one-dimensional representation \begin{align*} V_p(t) &= \int_0^t \int_{\{\theta\in U_pM:c(\theta)>s\}} J(s,\theta)\,d\sigma_p(\theta)\,d\mathcal{L}^1(s) \\ &= \int_0^t b(s)\,d\mathcal{L}^1(s). \end{align*} Therefore the fundamental theorem for locally absolutely continuous functions implies that $V_p$ is locally absolutely continuous on $I_k$ and \begin{align*} V_p'(t)=b(t) \end{align*} for $\mathcal{L}^1$-almost every $t\in I_k$. Next apply the standard coarea formula to the distance function $\rho:M\to[0,\infty)$, $q\mapsto d_g(p,q)$. The map $\rho$ is $1$-Lipschitz, and $|\nabla \rho|=1$ for $\operatorname{Vol}_g$-almost every point of $M\setminus\{p\}$ by the standard differentiability theorem for Lipschitz functions on Riemannian manifolds and the fact that the cut locus has zero Riemannian volume. Therefore, for every compact interval $[a,b]\subset I_k$ with $0<a<b$, \begin{align*} V_p(b)-V_p(a) &= \int_{\rho^{-1}((a,b))} 1\, d\operatorname{Vol}_g(q) \\ &= \int_a^b \mathcal{H}^{n-1}(S(p,t))\, d\mathcal{L}^1(t). \end{align*} It follows that $t\mapsto \mathcal{H}^{n-1}(S(p,t))$ is locally integrable on $I_k$ and, after changing neither function outside a null set nor the derivative, the fundamental theorem for locally absolutely continuous functions gives \begin{align*} V_p'(t)=\mathcal{H}^{n-1}(S(p,t)) \end{align*} for $\mathcal{L}^1$-almost every $t\in I_k$. Let $E\subset I_k$ be the intersection of the two full-measure sets on which the preceding two derivative identities hold. Then $E$ has full $\mathcal{L}^1$-measure in $I_k$, and for every $t\in E$, \begin{align*} b(t)=V_p'(t)=\mathcal{H}^{n-1}(S(p,t)). \end{align*} Thus it remains to prove that the ratio \begin{align*} t &\mapsto \frac{b(t)}{A_k(t)} \end{align*} is nonincreasing on $E$. [/step]

[step:Integrate the radial monotonicity over the surviving directions] Let $r,R \in E$ with $0<r<R$. Since $\{\theta \in U_pM : c(\theta)>R\} \subset \{\theta \in U_pM : c(\theta)>r\}$, the radial monotonicity from the previous step gives, for every $\theta$ with $c(\theta)>R$, \begin{align*} \frac{J(R,\theta)}{s_k(R)^{n-1}} \leq \frac{J(r,\theta)}{s_k(r)^{n-1}}. \end{align*} Integrating this inequality over the measurable set $\{\theta \in U_pM : c(\theta)>R\}$ with respect to $\sigma_p$ yields \begin{align*} \frac{1}{s_k(R)^{n-1}} \int_{\{\theta : c(\theta)>R\}} J(R,\theta)\, d\sigma_p(\theta) \leq \frac{1}{s_k(r)^{n-1}} \int_{\{\theta : c(\theta)>R\}} J(r,\theta)\, d\sigma_p(\theta). \end{align*} Because $J(r,\theta)\geq 0$ and $\{\theta:c(\theta)>R\}\subset\{\theta:c(\theta)>r\}$, enlarging the domain of integration on the right gives \begin{align*} \frac{b(R)}{s_k(R)^{n-1}} \leq \frac{1}{s_k(r)^{n-1}} \int_{\{\theta : c(\theta)>r\}} J(r,\theta)\, d\sigma_p(\theta) = \frac{b(r)}{s_k(r)^{n-1}}. \end{align*} Since \begin{align*} A_k(t)=\mathcal{H}^{n-1}(\mathbb{S}^{n-1})\,s_k(t)^{n-1}, \end{align*} division by the positive constant $\mathcal{H}^{n-1}(\mathbb{S}^{n-1})$ gives \begin{align*} \frac{b(R)}{A_k(R)} \leq \frac{b(r)}{A_k(r)}. \end{align*} [/step]

[step:Replace the derivative of volume by distance-sphere area] For $r,R \in E$, the defining property of $E$ gives \begin{align*} b(R)=\mathcal{H}^{n-1}(S(p,R)), \qquad b(r)=\mathcal{H}^{n-1}(S(p,r)). \end{align*} Substituting these identities into the preceding inequality gives \begin{align*} \frac{\mathcal{H}^{n-1}(S(p,R))}{A_k(R)} \leq \frac{\mathcal{H}^{n-1}(S(p,r))}{A_k(r)}. \end{align*} Thus the area ratio has a nonincreasing representative on the full-measure set of admissible radii, which is the asserted Bishop-Gromov area comparison. [/step]