Let $P_ℓ$ denote the space of complex-valued homogeneous polynomials of degree $ℓ$ in $n$ real variables, here considered as functions ${\displaystyle \mathbb {R} ^{n}\to \mathbb {C} }$.
Let $A_ℓ$ denote the subspace of $P_ℓ$ consisting of all harmonic polynomials:
$${\displaystyle \mathbf {A} _{\ell }:=\{p\in \mathbf {P} _{\ell }\,\mid \,\Delta p=0\}\,.}$$
These are the (regular) solid spherical harmonics. Let $H_ℓ$ denote the space of functions on the unit sphere
$${\displaystyle S^{n-1}:=\{\mathbf {x} \in \mathbb {R} ^{n}\,\mid \,\left|x\right|=1\}}$$
obtained by restriction from $A_ℓ$
$${\displaystyle \mathbf {H} _{\ell }:=\left\{f:S^{n-1}\to \mathbb {C} \,\mid \,{\text{ for some }}p\in \mathbf {A} _{\ell },\,f(\mathbf {x} )=p(\mathbf {x} ){\text{ for all }}\mathbf {x} \in S^{n-1}\right\}.}$$
Prove that for all $f \in H_ℓ$, one has
$${\displaystyle \Delta _{S^{n-1}}f=-\ell (\ell +n-2)f.}$$
where $\Delta _{S^{n-1}}$ is the Laplace–Beltrami operator on $S^{n-1}$. (This statement can be found on Wikipedia)
Best Answer
Let $p \in \mathbb{C}[x_1,\dots, x_n]$ be harmonic, homogenous of degree l. Let $X=\frac{x}{|x|}$ and $r=|x|$, thus $x=rX$, with $X \in S^{n-1}$. Then $p(x)=p(rX)=r^lp(X)$. Since $p$ Is harmonic it follows that:
$$ 0 = \Delta p=(\partial_r^2+\frac{n-1}{r}\partial_r+\frac{1}{r^2}\Delta_{S^{n-1}})(p)=l(l-1)r^{l-2}p(X)+\frac{n-1}{r}lr^{l-1}p(X)+\frac{1}{r^2}r^l\Delta_{S^{n-1}}p(X) $$
$$ \Leftrightarrow \Delta_{S^{n-1}}p(X)=(-l(l-1)-(n-1)l)p(X)=-l(l+n-2)p(X)$$
Therefore we find the eigenfunctions of $\Delta_{S^{n-1}}$ with eigenvalue $-l(l+n-2)$ by restricting the homogenous, harmonic polynomials of degree l to $S^{n-1}$
All credits to my mathematical physics prof :)