2016/03/20 by Lorenzo Foscolo, Foscolo, Lorenzo · 2 citations
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research
paper · pdf · doi:10.48550/arxiv.1603.06315
We construct large families of new collapsing hyperk "ahler metrics on the K3\nsurface. The limit space is the quotient of a flat 3-torus by an involution.\nAway from finitely many exceptional points the collapse occurs with bounded\ncurvature. There are at most 24 exceptional points where the curvature\nconcentrates, which always contains the 8 fixed points of the involution on the\n3-torus. The geometry around these points is modelled by ALF gravitational\ninstantons: of dihedral type (Dk) for the fixed points of the involution on the\n3-torus and of cyclic type (Ak) otherwise. The collapsing metrics are\nconstructed by deforming approximately hyperk "ahler metrics obtained by gluing\nALF gravitational instantons to a background (incomplete) hyperk "ahler metric\narising from the Gibbons-Hawking ansatz over a punctured 3-torus. As an\nimmediate application to submanifold geometry, we exhibit hyperk "ahler metrics\non the K3 surface that admit a strictly stable minimal sphere which cannot be\nholomorphic with respect to any complex structure compatible with the metric.\n