# Subelliptic estimates from Gromov hyperbolicity

## TL;DR

This paper establishes a connection between Gromov hyperbolicity of Kähler-Einstein metrics on convex domains and subelliptic estimates for the $ar{ ext{d}}$-Neumann problem, providing new characterizations and examples.

## Contribution

It proves that Gromov hyperbolicity implies subelliptic estimates for the $ar{ ext{d}}$-Neumann problem on convex domains without boundary regularity.

## Key findings

- Gromov hyperbolicity of the Kähler-Einstein metric implies subelliptic estimates.
- A new characterization of Gromov hyperbolicity via affine transformations.
- Examples where Gromov hyperbolicity of the Hilbert metric implies hyperbolicity of the Kähler-Einstein metric.

## Abstract

In this paper we prove: if the complete K\"ahler-Einstein metric on a bounded convex domain (with no boundary regularity assumptions) is Gromov hyperbolic, then the $\bar{\partial}$-Neumann problem satisfies a subelliptic estimate. This is accomplished by constructing bounded plurisubharmonic function whose Hessian grows at a certain rate (which implies a subelliptic estimate by work of Catlin and Straube). We also provide a characterization of Gromov hyperbolicity in terms of orbit of the domain under the group of affine transformations. This characterization allows us to construct many examples. For instance, if the Hilbert metric on a bounded convex domain is Gromov hyperbolic, then the K\"ahler-Einstein metric is as well.