# Extremal length in higher dimensions and complex systolic inequalities

## TL;DR

This paper extends the concept of extremal length to higher-dimensional complex manifolds and explores related geometric inequalities, including a complex analogue of the systole, to develop new tools for conformal invariants.

## Contribution

It introduces a higher-dimensional generalization of extremal length and formulates complex systolic inequalities, advancing the understanding of conformal invariants in complex geometry.

## Key findings

- Proposed a higher-dimensional extremal length framework
- Formulated complex systolic inequalities for moduli spaces
- Provided initial methods for estimating and computing the generalized extremal length

## Abstract

Extremal length is a classical tool in 1-dimensional complex analysis for building conformal invariants. We propose a higher-dimensional generalization for complex manifolds and provide some ideas on how to estimate and calculate it. We also show how to formulate certain natural geometric inequalities concerning moduli spaces in terms of a complex analogue of the classical Riemannian notion of systole.