The spherical harmonics are the eigenfunctions of the Laplace–Beltrami operator, the laplacian on functions, of the round sphere :
Each eigenvalue has multiplicity . In coordinates with the associated Legendre functions, and the real form replaces by or . They are the degree- homogeneous polynomials on annihilated by the Laplacian, restricted to the sphere, which is why is called the degree.
Two facts make them the right basis for geometry on the sphere:
Completeness. is a complete orthonormal basis of , so any field expands as . This is the Fourier analysis of the sphere.
Nodal structure. vanishes on parallels and meridians, cutting the sphere into a checker of sign domains; these nodal patterns are the sphere’s standing waves.
The mode is constant. It is the whole harmonic kernel on the sphere, which is simply connected, so and . Every higher mode carries a strictly positive eigenvalue, hence decays under the heat flow.