Minimal hypersurfaces in Kaehler scalar flat ALE spaces

TL;DR

This paper introduces a new method to analyze Kaehler scalar flat metrics on certain domains, enabling detection of minimal spheres, stability, and comparison of geometric inequalities in these spaces.

Contribution

A novel approach to describe Kaehler scalar flat metrics on U(n)-invariant domains, allowing for the detection of minimal spheres and stability analysis.

Findings

Penrose Inequality holds for these manifolds in all dimensions.

Stable minimal surfaces and divisors do not coexist in these spaces.

The new method simplifies the detection of minimal spheres and stability.

Abstract

In this paper we give a new general method to describe all Kaehler scalar flat metrics on $U(n)$-invariant domains of C^n in a way to be able to detect easily whether it can be completed to larger domains and which kind of ends they can have. This new approach makes possible to decide whether the corresponding metrics contain a minimal sphere among the standard euclidean ones. We will show also how to check the stability (i.e. minimizing volume up to second order) of such submanifolds. We apply the present analysis to a comparison between Penrose Inequality, which requires the presence of a stable minimal hypersurface, and Hein-LeBrun's, which requires the ambient space to be Kaehler and predicts the existence of a special divisor, and we will show that Penrose Inequality does indeed hold in any dimension for this class of manifolds, and that the two inequalities are indeed incomparable in that for these manifolds a stable minimal surface and a divisor never coexist.