Let $S$ be the round $n$-sphere of radius $R$ in Euclidean space, and let $r$ be the intrinsic distance from the north pole. Further, let $U(r)$ be the spherical cap of intrinsic radius r. (So $U(0)$ is the north pole and $U(\frac{\pi R}{2})$ is the upper hemisphere.)
Let $\lambda(r)$ be the first eigenvalue of the Laplacian on $U(r)$, with Dirichlet boundary condition. What is the expression of $\lambda(r)$ ?
For $n=3$, $$\lambda(r)=\frac{\pi^2}{r^2}-1,\ 0<r\leq\pi.$$
Unfortunately, the first Dirichlet eigenvalue for spherical caps in other dimensions cannot be calculated explicitly, but some estimates are known: $$\lambda(r)=\frac{j_{(n-2)/2,1}^2}{r^2}+O(1)\quad\text{as } r\to0^+,$$ where $j_{\nu,1}$ is the first zero of Bessel function $j_\nu$, and as $r\to\pi^{-}$, $$\lambda(r)=\begin{cases}c_{n}\left(\pi-r\right)^{n-2}+o\left(\left(\pi-r\right)^{n-2}\right),\quad & n\geq3;\\ -c_{2}\log^{-1}(\pi-r)+o\left(\log(\pi-r)^{-1}\right),\quad &n=2. \end{cases}$$ for some constants $c_n$. The proof and more sharper estimates on $\lambda(r)$ can be found in a paper by Borisov and Freitas.
