Laplace spectrum of the $2$-Sphere The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this operator must be very well known. Where can one see these values written out explicitly?
 A: There is a very simple way to compute the Laplace-Beltrami $\Delta_S$ operator of a function on the sphere $f:\mathbb{S}^{n-1}\to\mathbb{C}$. You simply extend the function to $\mathbb{R}^n\setminus\{0\}$ by $F(x)=f(x/|x|)$ and define
$$
\Delta_Sf(x)=(\Delta F)|_{\mathbb{S}^{n-1}},\quad
\text{ where   }  \quad \Delta F=-\sum_{i=1}^n F_{x_ix_i}.
$$
Let $\mathcal{P}_k$ be the space of all homogeneous polynomials of degree $k$, $P(x)=\sum_{|\alpha|=k}a_\alpha x^\alpha$. The elements of the subspace $\mathcal{H}_k\subset\mathcal{P}_k$ consisting of polynomials that are harmonic functions are called  solid spherical harmonics. Then restrictions of solid spherical harmonics to $\mathbb{S}^{n-1}$ are called surface spherical harmonics. The space of surface spherical harmonics is denotes by $H_k$.

Theorem. The subspaces $H_k\subset L^2(\mathbb{S}^{n-1})$ are mutually orthogonal and
  $$
L^2(\mathbb{S}^{n-1})=H_0\oplus H_1\oplus H_2\oplus\ldots
$$
  Moreover
  $$
\Delta_S Y(x)=k(k+n-2)Y \quad\text{ for $Y\in H_k$}
$$

In your case $n=3$ so the eigenvalues are $k(k+1)$.
There are many places where you can find a proof of this result. See for example page 67 in my notes. Note that in my notes I defined the Laplace operator without the minus sign so the eigenvalues are $-k(k+n-2)$.
P. Hajlasz, Functional Analysis. Available at:
http://www.google.com/url?q=http%3A%2F%2Fwww.pitt.edu%2F~hajlasz%2FNotatki%2FFunctional%2520Analysis2.pdf&sa=D&sntz=1&usg=AFQjCNEgjTnVgLE_daoXzgSa6ggIkXuiXA
