Let $(M,g)$ be a complete $n$-dimensional Riemannian manifold and let $p \in M$. Consider $(t,\Theta)$ , the geodesic spherical coordinates around $p$, where $t \in (0,\text{conj}_p(\Theta))$ and $\Theta$ is a unit vector in $T_pM$. Let $A_p(t,\Theta)$ be the density of the volume measure in these coordinates, i.e.
\begin{equation*}
d\operatorname{Vol} = A_p(t,\Theta) dt d\Theta
\end{equation*}
A well-known theorem of Gromov states that if $\operatorname{Ric}(M) \geqslant (n-1)\kappa$, then the map
\begin{equation}
t \mapsto \frac{{A}_p(t,\Theta)}{sn^{n-1}_{\kappa}(t)}
\end{equation}
is non-increasing in $t$. As usual, $sn_{\kappa}$ is given by
\begin{align*}
sn_{\kappa}(t) = \begin{cases}
\frac{\sin{\sqrt{k}t}}{\sqrt{k}} & k > 0\\
t & k = 0\\
\frac{\sinh{\sqrt{-k}t}}{\sqrt{-k}} & k < 0
\end{cases}
\end{align*}
Now I would like to prove a similar result when the sectional curvature of $M$ is bounded from above. That is, if $ \text{sec}(M) \leqslant \kappa$, then
\begin{equation*}
\frac{d^2}{dt^2}\left(\frac{A_p(t,\Theta)}{sn^{n-2}_{\kappa}(t)}\right) + \kappa \left(\frac{A_p(t,\Theta)}{sn^{n-2}_{\kappa}(t)}\right) \geqslant 0
\end{equation*}
I'm trying to mimic the argument given by Gromov, letting $\varphi(t) = A_p(t,\Theta)^{\frac{1}{n-2}}$ and calculate that $(\log \varphi(t))' = \frac{1}{n-2}\text{tr}(\text{II}(t))$, where $\text{II}(t)$ is the second fundamental form of $\partial B(p,t)$ . But since we are not proving a statement about monotonicity, I don't know how I can get rid of the power $(n-2)$. Differentiating such expression directly seems intimidating and tedious, and I believe there's a shortcut to the problem since it is very similar to the estimate of the norm of Jacobi fields. Any insight of the problem will be appreciated.
Best Answer
So my professor gave me an idea of how to solve this problem. After we get \begin{equation*} \underbrace{[\text{tr}(\text{II}(t))-(n-2)\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}]'}_{\text{Part A}} + \underbrace{[\text{tr}(\text{II}(t))-(n-2)\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}]^2}_{\text{Part B}} + \kappa \geqslant 0 \tag{$\star$} \end{equation*} We can use Riccati's equation to rewrite \begin{align*} \text{Part A} = &[\text{tr}(\text{II}(t))-(n-2)\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}]' \\ \geqslant & -\text{tr}(\text{II}(t)^2)-(n-1)\kappa - (n-2)[-(\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)})^{2} -\kappa]\\ =& -\text{tr}(\text{II}(t)^2) + (n-2)(\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)})^{2}-\kappa \end{align*} And after expanding out $\text{Part B}$, $\star$ becomes \begin{align*} &\underbrace{[\text{tr}(\text{II}(t))-(n-2)\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}]'}_{\text{Part A}} + \underbrace{[\text{tr}(\text{II}(t))-(n-2)\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}]^2}_{\text{Part B}} + \kappa \\ \geqslant & \text{tr}(\text{II}(t))^2-\text{tr}(\text{II}(t)^2) -2(n-2)\text{tr}(\text{II}(t))\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}' + (n-1)(n-2)(\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)})^2\\ \geqslant & \text{tr}(\text{II}(t))^2-\text{tr}(\text{II}(t)^2) -2(n-2)\text{tr}(\text{II}(t))\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}' + (\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)})^2\\ = &\sum_{1,i\neq j,n-1}\lambda_{i}(t)\lambda_j(t) - [\lambda_{i}(t)+\lambda_{j}(t)]\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}' + (\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)})^2\\ =& \sum_{1,i\neq j,n-1}(\lambda_i(t)-\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}')(\lambda_j(t)-\frac{\text{sn}'_{\kappa}(t)}{\text{sn}_{\kappa}(t)}')\\ \geqslant & 0 \end{align*} where $\lambda_{i}(t), i=1,\dots,n-1$ are the eigenvalues of $\text{II}(t)$. The last inequality follows from Hessian comparison, which is indicated in Corollary 2.4 in Petersen's book.