Log-Concavity of First Dirichlet Eigenfunctions on ${\mathbb{C}\mathbb{P}}^2$

We study the log-concavity property of first Dirichlet eigenfunctions on domains in ${\mathbb{C}\mathbb{P}}^2$. For every smooth $1$-convex domain $\mathrm{\Omega }\subset {\mathbb{C}\mathbb{P}}^2$, we prove the quantitative estimate $${\mathrm{\nabla }}^2(-\log u)>\max \{\psi (s),\frac{8}{5}\}g,\text{ where }\psi (\mathrm{\mid }\text{grad}f{\mathrm{\mid }}^2)=\frac{s}{\sqrt{1+s}}-\log (1+s),$$ for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound ${\lambda }_2-{\lambda }_1>46\mathrm{/}5$.