Fundamental gaps of spherical triangles

We compute Dirichlet eigenvalues and eigenfunctions explicitly for spherical lunes and the spherical triangles which are half the lunes, and show that the fundamental gap goes to infinity when the angle of the lune goes to zero. Then we show the spherical equilateral triangle of diameter $\frac{\pi}{2}$ is a strict local minimizer of the fundamental gap on the space of the spherical triangles with diameter $\frac{\pi}{2}$, which partially extends Lu-Rowlett's result from the plane to the sphere.