A remark on the Herzlich volume of asymptotically complex hyperbolic Einstein manifolds

TL;DR

This paper investigates inequalities relating the Herzlich volume and Euler characteristic of 4D asymptotically complex hyperbolic Einstein manifolds, focusing on cases where the metric is Kaehler or selfdual, with additional assumptions in the selfdual case.

Contribution

It establishes new inequalities connecting the Herzlich volume and topological invariants for specific classes of Einstein manifolds, under certain geometric conditions.

Findings

Inequalities involving Herzlich volume and Euler characteristic for Kaehler cases.

Additional inequalities for selfdual cases with non-vanishing Kronheimer-Mrowka invariant.

Conditions under which these inequalities hold for asymptotically complex hyperbolic Einstein manifolds.

Abstract

We observe inequalities involving the Herzlich volume of a 4-dimensional asymptotically complex hyperbolic Einstein manifold and its Euler characteristic provided the metrics is either Kaehler or selfdual. In the selfdual case we have to assume furthermore that the Kronheimer-Mrowka invariant is non vanishing.