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We derive a sufficient condition on a bounded pseudoconvex domain$\Omega\subset\mathbb{C}^2$ with smooth boundary such that $-(-\rho)^\eta$ isplurisubharmonic on $\Omega$ for $\eta>0$ arbitrarily close to $1$ (thesupremum 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 Herbigin 2007 and only requires restriction on Levi-flat sets of the boundary$\partial\Omega$. Since the condition is on Levi-flat sets, it contains moregeometric information. As an application of this new condition, we discuss howthe 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 $1$ ifonly the Levi-flat sets form a real curve transversal to the holomorphictangent vector fields on $\partial\Omega$ (see Theorem [main]). We also give aspecific example (see Theorem [example]) on the bounded pseudoconvex domainswhich verify the application but are neither of finite type nor admit aplurisubharmonic defining function on the boundary.

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