Geometric Analysis on the Diederich-Forn{\ae}ss Index

We derive a sufficient condition on a bounded pseudoconvex domain $\Omega\subset\mathbb{C}^2$ with smooth boundary such that $-(-\rho)^\eta$ is plurisubharmonic on $\Omega$ for $\eta>0$ arbitrarily close to $1$ (the supremum of $\eta$ is called Diederich-Forn{\ae}ss index, see Definition (df)). This condition (see Theorem prop) extends a theorem of Forn{\ae}ss and Herbig in 2007 and only requires restriction on Levi-flat sets of the boundary $\partial\Omega$. Since the condition is on Levi-flat sets, it contains more geometric information. As an application of this new condition, we discuss how the geometry of the Levi-flat sets affects the Diederich-Forn{\ae}ss index. Among other results, we show that the Diederich-Forn{\ae}ss index is $1$ if only the Levi-flat sets form a real curve transversal to the holomorphic tangent vector fields on $\partial\Omega$ (see Theorem [main]). We also give a specific example (see Theorem [example]) on the bounded pseudoconvex domains which verify the application but are neither of finite type nor admit a plurisubharmonic defining function on the boundary.