We construct large families of new collapsing hyperk\"ahler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which always contains the 8 fixed points of the involution on the 3-torus. The geometry around these points is modelled by ALF gravitational instantons: of dihedral type (Dk) for the fixed points of the involution on the 3-torus and of cyclic type (Ak) otherwise. The collapsing metrics are constructed by deforming approximately hyperk\"ahler metrics obtained by gluing ALF gravitational instantons to a background (incomplete) hyperk\"ahler metric arising from the Gibbons-Hawking ansatz over a punctured 3-torus. As an immediate application to submanifold geometry, we exhibit hyperk\"ahler metrics on the K3 surface that admit a strictly stable minimal sphere which cannot be holomorphic with respect to any complex structure compatible with the metric.

中文翻译：

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$K3$ 表面上的 ALF 引力瞬子和塌陷 Ricci-flat 度量

我们在 K3 表面上构建了大量新的塌陷 hyperk\"ahler 度量。极限空间是平面 3 环面与对合的商。远离有限的许多异常点，塌陷以有界曲率发生。至多有曲率集中的 24 个异常点，其中始终包含 3 环面上对合的 8 个不动点。这些点周围的几何由 ALF 引力瞬子建模：二面体类型 (Dk) 用于对合的不动点3-圆环和循环类型 (Ak) 否则。通过将 ALF 引力瞬时子粘合到背景（不完整）hyperk\"ahler 度量而获得的近似 hyperk\"ahler 度量变形来构建崩溃度量，该度量由 Gibbons-Hawking ansatz 产生在一个刺破的 3 圆环上。作为对子流形几何的直接应用，我们在 K3 表面上展示了 hyperk\"ahler 度量，该度量允许严格稳定的最小球体，对于与该度量兼容的任何复杂结构，该最小球体不能是全纯的。