A gluing construction of $D_{k}$ ALF gravitational instantons and existence of non-holomorphic minimal spheres

TL;DR

This paper extends the construction of $D_k$ ALF gravitational instantons by combining the $D_1$ Atiyah--Hitchin metric with multiple Taub-NUT metrics, and shows these spaces contain non-holomorphic minimal spheres, contributing to the understanding of $K3$ surfaces.

Contribution

It introduces a new gluing construction for $D_k$ ALF gravitational instantons involving the $D_1$ Atiyah--Hitchin metric and establishes the existence of non-holomorphic minimal spheres in these spaces.

Findings

Constructed new $D_k$ ALF gravitational instantons with mixed superpositions.

Proved the existence of non-holomorphic minimal spheres in these spaces.

Identified large classes of $K3$ surfaces with non-holomorphic minimal spheres.

Abstract

This note extends the construction of $D_{k}$ ALF gravitational instantons in Schroers--Singer to a new case where the nonlinear superposition is given by the $D_{1}$ Atiyah--Hitchin metric and $k-1$ copies of $A_{0}$ Taub-NUT metrics. We then give a general class of ALF spaces such that each of them contains a non-holomorphic minimal sphere. Together with Foscolo's construction this gives a large class of $K3$ surfaces containing non-holomorphic minimal spheres.