The Strong Diederich-Forn{\ae}ss Index on $C^2$ Domains in Hermitian Manifolds

TL;DR

This paper characterizes the strong Diederich-Forn{ a}ss index for $C^2$ domains in Hermitian manifolds through curvature conditions of a Hermitian metric, linking geometric properties to complex Hessian bounds.

Contribution

It provides a complete characterization of the strong Diederich-Forn{ a}ss index using curvature inequalities on Hermitian manifolds, extending understanding of boundary regularity.

Findings

The strong Diederich-Forn{ a}ss index is characterized by curvature conditions.

Existence of a Hermitian metric with specific curvature bounds is equivalent to the index.

The results connect boundary geometry with complex Hessian estimates.

Abstract

For a relatively compact Stein domain $\Omega$ with $C^2$ boundary in a Hermitian manifold $M$, we consider the strong Diederich-Forn{\ae}ss index, denoted $DF(\Omega)$: the supremum of all exponents $0<\eta<1$ such that eigenvalues of the complex Hessian of $-(-\rho)^\eta$ are bounded below by some positive multiple of $(-\rho)^\eta$ on $\Omega$ for some $C^2$ defining function $\rho$. We will show that $DF(\Omega)$ is completely characterized by the existence of a Hermitian metric with curvature terms satisfying a certain inequality when restricted to the null-space of the Levi-form.