I want to find explicit formulas for the eigenfunctions of the Laplacian on $\mathbb{CP}^n$ endowed with the Fubini Study metric.

For the first eigenvalue $\lambda_1 = 4(n+1)$, the eigenfunctions are given by the real and imaginary parts of $\phi^{i, j} = \frac{z_i\bar{z_j}}{\sum_{k}|z_k|^2}-\frac{\delta_{i, j}}{n+1}$, and these functions together gives the Veronese isometric embedding of $\mathbb{CP}^n$ into $S^{2(n+1)^2-1}$.

Is there an analogue of this for higher eigenfunctions? Is the $k$-th eigenspace also associated with some embedding of $\mathbb{CP}^n$? Are there explicit formulas listing the higher eigenfunctions like the $\phi$ above?

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    $\begingroup$ Duplicate ? mathoverflow.net/questions/114107/… $\endgroup$ – Alexander Chervov Apr 6 '20 at 22:18
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    $\begingroup$ Not duplicate. That question asked for the $\lambda_1$ eigenspace; the new question gives the $\lambda_1$ eigenspace and asks for the $\lambda_k$ eigenfunctions for $k>1$. $\endgroup$ – Noam D. Elkies Apr 6 '20 at 22:39
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    $\begingroup$ Note, by the way, that your claimed formula for the eigenfunctions belonging to $\lambda_1$ on $\mathbb{CP}^n$ is not quite correct. For example, your $\phi^{1,1}$ is real-valued and non-negative, so its integral over $\mathbb{CP}^n$ can't be zero, which would be so for an actual eigenfunction belonging to $\lambda_1$. You are off by an additive constant that you need to subtract in order to get the average value of the function to be $0$. $\endgroup$ – Robert Bryant Apr 7 '20 at 11:08
  • $\begingroup$ Yes, I made an edit. $\endgroup$ – freidtchy Apr 7 '20 at 16:09

The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$.

Now, for a given $k\ge0$, let $H_k$ be the (real) vector space of real-valued polynomials $p(z,\bar z)$ that are homogeneous of degree $k$ in the $z$-variables and degree $k$ in the $\bar z$-variables and that are harmonic, i.e., they satisfy $$ \frac{\partial^2p}{\partial z_0\partial \bar z_0}+\cdots+ \frac{\partial^2p}{\partial z_n\partial \bar z_n} = 0. $$ One easily computes that $$ \dim_\mathbb{R}H_k = {{n+k}\choose{k}}^2 -{{n+k-1}\choose{k-1}}^2. $$

Then, for $p\in H_k$, the function $f_p:\mathbb{CP}^n\to\mathbb{R}$ given by $$ f_p\bigl([z]\bigr) = \frac{p(z,\bar z)}{Z^k} $$
is well-defined and is an eigenfunction with eigenvalue $\lambda_k$. Conversely, every eigenfunction with eigenvalue $\lambda_k$ is of this form for a unique $p\in H_k$.

Added remark: I was privately asked how to prove that the description I gave of the eigenvalues and eigenfunctions of $\mathbb{CP}^n$ is correct. I'm sure it's in the literature in various places, but it's easier to just give the argument, using the known description of the eigenvalues and eigenfunctions on the standard sphere. Here's the idea:

Let $S^{2n+1}\subset \mathbb{C}^{n+1}$ be the standard unit $2n{+}1$-sphere with its induced metric and let $\sigma:S^{2n+1}\to\mathbb{CP}^n$ be the quotient mapping $\sigma(z) = [z] = \mathbb{C}z$. Then $\sigma$ is a Riemannian submersion when $\mathbb{CP}^n$ is given the Fubini-Study metric (appropriately scaled; with this choice, the area of a linear $\mathbb{CP}^1\subset\mathbb{CP}^n$ is $\pi$, not $4\pi$).

If $f$ is an eigenfunction on $\mathbb{CP}^n$ with eigenvalue $\lambda_k$, then $\sigma^*(f)$ is an eigenfunction on $S^{2n+1}$ with eigenvalue $\lambda_k$, one that is, moreover, invariant under the circle action $\mathrm{e}^{i\phi}\cdot z = \mathrm{e}^{i\phi}z$ on $S^{2n+1}$.

Now, it is known that, if $\mu_d$ is the $d$-th eigenvalue of $S^{2n+1}$, then any eigenfunction $f$ with this eigenvalue is the restriction to $S^{2n+1}$ of a harmonic polynomial on $\mathbb{R}^{2n+2}$ that is homogeneous of degree $d$. We are looking for such polynomials that are also invariant under the circle action of multiplication by $\mathrm{e}^{i\phi}$. When expressed as a polynomial in the complex coordinates $z_i$ and $\bar z_i$, a polynomial that is invariant under this circle action must have the same number of $z$'s as $\bar z$'s. This can only happen if $d$ is even. Conversely, if $d=2k$, then we get the space $H_k$ as defined above. Thus, $\lambda_k = \mu_{2k}$, and the above description is justified.

  • $\begingroup$ Thanks for the answer. Do you know if the higher eigenfunctions arises from some embedding of $CP^n$ like it does for the first eigenfunction? $\endgroup$ – freidtchy Apr 7 '20 at 19:46
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    $\begingroup$ @freidtchy: Yes. Think of the mapping $p\mapsto f_p\bigl([z]\bigr)\in\mathbb{R}$ for a given $[z]\in\mathbb{CP}^n$ as a linear function on $H_k$, so this defines a map of $\mathbb{CP}^n$ into ${H_k}^*$ (the dual space of $H_k$). This is an embedding of $\mathbb{CP}^n$ into ${H_k}^*$, which, when $k=1$, is the one you already know (since $H_1$ is essentially the traceless Hermitian forms on $\mathbb{C}^{n+1}$). This mapping is equivariant with respect to $\mathrm{SU}(n{+}1)$, as $H_k$ is an irreducible representation of this group (up to a covering, the isometry group of $\mathbb{CP}^n$). $\endgroup$ – Robert Bryant Apr 7 '20 at 21:06

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