What question did this study set out to answer?

The aim is to extend the construction of ALF gravitational instantons to a new case involving the Atiyah-Hitchin metric.

February 12, 2026

A gluing construction of 𝐷👷 ALF gravitational instantons and existence of non-holomorphic minimal spheres

Key Points

The aim is to extend the construction of ALF gravitational instantons to a new case involving the Atiyah-Hitchin metric.
Extended the construction of Dk ALF gravitational instantons based on previous works.
Introduced a non-linear superposition using the D1 Atiyah-Hitchin metric.
Combined k-1 copies of A0 Taub-NUT metrics to form a larger class of ALF spaces.
Identified a general class of ALF spaces containing non-holomorphic minimal spheres.
Established a connection with Foscolo’s construction, enriching the class of K3 surfaces that contain non-holomorphic minimal spheres.

Abstract

This note extends the construction of D k D₊ ALF gravitational instantons in Schroers–Singer Q. J. Math. 72 (2021), pp. 277–337 to a new case where the non-linear superposition is given by the D 1 D₁ Atiyah–Hitchin metric and k − 1 k-1 copies of A 0 A₀ 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 J. Differential Geom. 112 (2019), pp. 79–120 this gives a large class of K 3 K3 surfaces containing non-holomorphic minimal spheres.

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