Acta Mathematica

Volume 232 (2024)

Number 2

Collapsing geometry of hyperk\”ahler 4-manifolds and applications

Pages: 325 – 424

Authors

Song Sun (Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou, China and Department of Mathematics, University of California, Berkeley, Berkeley, CA, U.S.A.)

Ruobing Zhang (Department of Mathematics, University of Wisconsin–Madison, Madison, WI, U.S.A. and Department of Mathematics, Princeton University, Princeton, NJ, U.S.A.)

Abstract

We investigate the collapsing geometry of hyperk\"ahler $4$-manifolds. As applications, we prove the following two well-known conjectures in the field.

(1) Any collapsed limit of unit-diameter hyperk\"ahler metrics on the K3 manifold is isometric to one of the following: the quotient of a flat $3$-torus by an involution, a singular special K\"ahler metric on the $2$-sphere, or the unit interval.

(2) Any complete hyperk\"ahler $4$-manifold with finite energy (i.e., gravitational instanton) is asymptotic to a model end at infinity.

The full text of this article has not yet been published online.

The first author is supported by the Simons Collaboration on Special Holonomy in Geometry,Analysis and Physics (# 488633), and NSF grant DMS-2004261. The second author is supported byNSF grant DMS-1906265.

Received 29 September 2021

Accepted 22 December 2022

Published 8 October 2024