Strict Log-Concavity of First Dirichlet Eigenfunctions under Symmetry in ${\mathbb{C}\mathbb{P}}^n$

We prove that the positive first Dirichlet eigenfunction of every smooth domain with convex boundary in ${\mathbb{C}\mathbb{P}}^n$ invariant under the standard $U(1)\times U(n-1)$-action is strictly log-concave for $n=2,3,4$.