Extremal length in higher dimensions and complex systolic inequalities

Understand

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.