For the 2-sphere $\mathbb{S}^2$, the first eigenvalue of the spherical cap can be calculated via stereographic projection. Under this projection, $U(r)$ is a ball $B_{\mathbb{C}}(0,\tan(r/2))$ in $\mathbb{C}$, whose first eigenvalue is $$\lambda_{1}(r)=\left(\frac{\mu_1}{\tan(r/2)}\right)^{2},$$ where $\mu_1$ is the fist zero of the Bessel function $$J_{0}(t)=\frac{1}{\pi}\int_{0}^{\pi}\cos(t\sin(\theta))\ \mathrm{d}\theta$$ and $\mu_1\approx2.4048$.
A. Chen
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