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 $\mu_1\approx2.4048$.