Does the first Laplacian eigenfunction on a homogeneous space have a unique maximum?

Question

For convex domains $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, it's known that any first Laplacian eigenfunction is log-concave. In particular, it has a unique maximum.

These are functions $f$ satisfying $-\Delta f = \lambda_1 f$ for the smallest possible $\lambda_1 > 0$ and $f|_{\partial \Omega} = 0$.

Q: Is there such a concavity result for homogeneous spaces, ie. Riemannian manifolds with a transitive action of the isometry group? Does $f$ at least have a unique maximum?

I imagine that this might even be apparent from an explicit formula for these eigenfunctions, but I haven't been able to find one in general.

I can at least check that it holds for spherical harmonics. Actually for spheres I think it can be derived from the domain result by separation of variables (take $\Omega = \mathbb{R}^n$). Perhaps such a method can work for any homogeneous space with a nice enough embedding, such as as the boundary of a convex domain?

I'd appreciate any thoughts or relevant references!

Answer 1

The flat torus $\mathbb{T} = \mathbb{R}^2/\Lambda$ gives a counterexample: The first nontrivial eigenvalue is of the form $\lambda_1 = \xi_1^2+\xi_2^2$, where $\xi = (\xi_1,\xi_2)$ is a nonzero element of the dual lattice $\Lambda^*$ of smallest norm, and the correspnding eigenfunctions are of the form $f(x_1,x_2) = a\cos(\xi_1 x_1 + \xi_2 x_2 + b)$ for some constants $(a,b)$. This function has a whole circle of maxima.

In general, on a compact manifold, an eigenfunction with nonzero eigenvalue must change sign because its average value on the manifold must be zero. (Integration by parts.)

Oh, another example occurred to me that you might find more interesting: Let $M = \mathrm{SO}(3)$ with its standard biïnvariant Riemannian metric. The first nontrivial eigenvalue has multiplicity 9, and the corresponding eigenfunctions are the 9 entries $a_{ij}$ of the standard matrix embedding of $\mathrm{SO}(3)$ into the space of 3-by-3 matrices. Each $a_{ij}$ has its maximum and minimum values equal to $\pm 1$, but it attains each on a circle embedded in $\mathrm{SO}(3)$. Meanwhile $f = -a_{11}-a_{22}-a_{33}$ has a maximum value of $1$, attained on a copy of $\mathbb{RP}^2$ embedded in $\mathrm{SO}(3)$, and a minimum value of $-3$, attained only at $I_3\in\mathrm{SO}(3)$.