# The geometry of $m$-hyperconvex domains

## TL;DR

This paper investigates the geometric properties of $m$-hyperconvex domains, establishing the existence of special exhaustion functions with desirable smoothness and subharmonicity properties within the Caffarelli-Nirenberg-Spruck framework.

## Contribution

It proves that every $m$-hyperconvex domain admits a negative, smooth, strictly $m$-subharmonic exhaustion function with bounded $m$-Hessian measure, advancing understanding of their geometric structure.

## Key findings

- Existence of special exhaustion functions for $m$-hyperconvex domains.
- Characterization of domain geometry via barrier functions and Jensen measures.
- Enhanced understanding of $m$-subharmonic functions in complex analysis.

## Abstract

We study the geometry of $m$-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every $m$-hyperconvex domain admits an exhaustion function that is negative, smooth, strictly $m$-subharmonic, and has bounded $m$-Hessian measure.